Structure of Arbitrary Purely Inseparable Extensions (Lecture Notes in Mathematics, 173) 354005295X, 9783540052951
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Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
173 John N. Mordeson Creighton University, Omaha, NB/USA
Bernard Vinograde Iowa State University, Ames, lA/USA
Structure of Arbitrary Purely Inseparable Extension Fields
Springer-Verlag Berlin· Heidelberg· NewYork 1970
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
173 John N. Mordeson Creighton University, Omaha, NB/USA
Bernard Vinograde Iowa State University, Ames, lA/USA
Structure of Arbitrary Purely Inseparable Extension Fields
Springer-Verlag Berlin· Heidelberg· NewYork 1970
3-540-05295-X Springer-Verlag Berlin' Heidelberg' New York ISBN 0-387-05295-X Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1970. Library of Congress Catalog Card Number 70-142789 Printed in Germany. Offsetdruck: Julius Beltz, Weinheim/Bergstr.
Preface Starting with O. Teichmuller's basic concepts [56J, G. Pickert developed an extensive theory of purely inseparable extensions, especially the finite degree case
[43J.
In these
Notes we present an infinite degree theory, especially for the case without exponent.
In addition to our own research, we
include many relevant results from other sources which are acknowledged in the Reference Notes following each chapter. We stop short of the emerging Galois theory but have listed a number of references that may be consulted.
It is assumed
that the reader is acquainted with the elements of purely inseparable extensions such as appear
in Jacobson
[24J.
Throughout these Notes L/K always denotes a field extension of a field K of characteristic p f O.
1sl means the cardinality
of a set S and C means proper containment.
Contents I.
II.
III.
Generators .
1
A.
Relative p-bases
B.
Extensions of type
C.
Special generating systems •.•.••..•.•••••••••••••
24
D.
Modular extensions
50
E.
Extens ion exponents ...••.•••••••••.•••••••••••••••
62
9
R
Intermediate fields A.
Lattice invariants
B.
More on type
.
.
R •••••••••••••••••••••••••••••••••••
74 86
Some applications
A.
Extension coefficient fields
B.
Fie Ld compos i tes
93 .
...................................... 73, 92, ....................................................
113
Reference Notes
134
References
135
I. A. Relative p-bases.
Generators
We collect here some old and some new
facts about p-bases that will be used frequently in our work. 1.1. of
L/K
sets B
Definition. is a subset
B'
of
B
of
K(LP,B ')
L
K(LP,B).
C
subset
Definition.
M of
L/K
and
L
When
K
is perfect,
M of
L.
L/K
is a
is relatively p-independent
When
K
is perfect,
M is the
L.
Definition.
is a subset
M of
subsets
of
M'
M
L = K(LP,M).
usual p-base of
1.3.
A relative p-base
such that
B
such that for all proper sub-
is the usual (absolutely) p-independent subset of 1.2.
in
B,
A relatively p-independent subset
M,
a minimal set over
L
A minimal generating set such that
K(M') K
L.
C
if
L
=
K(M)
A subset
M of
L/K
and for all proper M of
L
is called
M is a minimal generating set of
K(M)/K.
1.4. let
Proposition.
M be a subset of
set of
L/K
p-base of
L.
Then L
=
be purely inseparable and M
is a minimal generating
K(M)
and
M
is a relative
L/K. If
M
L = K(M)
is a minimal generating set of and thus
relatively p-independent in such that
L/K
if and only if
Proof. obviously
Let
m
L = K(LP,M) . L/K,
If
L/K,
M
then there exists
K(LP,M-m) = K(mP,M- m).
Thus
m
then
is not m c M
is both
2 purely inseparable and separable algebraic over whence
m
K(M-m).
However this contradicts the fact that
M is a minimal set over
K.
is a relative p-base of
L/K,
K(L P)
Conversely if then
whence a minimal set over 1.5.
K(M-m),
Definition.
L/K
M
L
and
K(M)
M
is a minimal set over q.e.d.
K.
is said to have an exponent (or,
to be of bounded exponent) if and only if there exists a nonnegative integer LP
e
=
raP
e
Ia
exponent of
e L}.
L/K.
LP
such that
e
K,
where
The smallest such integer is called the A purely inseparable extension without
exponent is also said to be of unbounded exponent.
(Note that
an extension with exponent must be purely inseparable.)
1.6.
Corollary.
be a subset of L/K
if and only if Proof.
L
L.
If
= K(LP,M)
M
whence
Let
Then
L/K
M
have exponent
e
is a relative p-base of
=
e
K(L P ,M)
= K(M).
L/K.
L/K,
then
Hence the conclusion
follows from Proposition 1.4. 1.7. and let
Proposition. e
(a)
Let
B UC
Band
C
be subsets of
L/K
Then
is relatively p-independent in
a minimal generating set of if
q.e.d.
be a positive integer. B
M
is a minimal generating set of
M is a relative p-base of
L
and let
L/K
K(L P) (B,C)/K(LP) (B)
is relatively p-independent in
L/K
and
C
is
if and and
B n C
=
¢.
3 If
(b) B n C == ¢, -e L(C P )/K.
B U cP
then
L/K
then
If
(c)
is relatively p-independent in L/K and -e B n cP is relatively p-independent in
B U C
B U C -e
is a relative p-base of
L/K and -e L(C P )/K.
is a relative p-base of
Proof.
(a)
and
is a minimal generating set of
C
Suppose
B
b e B
If there exists exists
c
C
E:
property,
c
such that
C
K(LP)(B,C)/K(LP)(B).
c e K(L P) (B,C - c).
such that
b e K(LP)(B-b,C),
such that b
t K(LP)(B-b,C-c).
K(L P) (B, C - c),
then there
By the exchange
which is impossible.
The converse
is immediate. (b) e == 1. b
E:
q.e.d.
By induction, it suffices to prove the theorem for If there exists
K«L(C P
-1
b e B
such that
))P)(B-b,C) == K(LP)(C,B-b),
the relative p-independence of relatively p-independent in
B U C
L(C P
-1
in
)/K.
then we contradict L/K.
Hence
B
is
We now apply part (a)
-1
by showing c P is a minimal generating set of -1 -1 -1 K«L(C P ))P)(B,CP )/K«L(CP ))P(B). Suppose there exists such that
cP
-1
c
C
c
KP ( LP (cP ) )( BP , C - c)
2
¢,
is relatively p-independent in
c e C
Then there does not exist
B n C ==
K(LP(C))(B,CP K( LP) ( C - c)
the relative p-independence of
B UC
-1
- cP
-1
).
Then
which again contradicts in
L/K.
4 (c) B U cP (b).
Suppose
-e
B U C
is a relative p-base of L(C P
is relatively p-independent in K((L(C P
Now
-e
))P)(B,C P
-e
-e
)/K -e
K(LP(C,B))(C P
)
L/K.
Then
by part -e
L(C P
)
).
q.e.d.
1.8.
Proposition.
Let
L'
a purely inseparable extension (L: L'l
L/K.
b
then since
exchange property that b
is relatively p-independent in
If there exists
K(L P) (M- m, b),
E:
M U (b}
is distinguished in
L/K,
L/K.
When
we say that
L/L'
if and is also
is perfect
L/L'
preserves
p-independence. Proposition.
1.10.
L/K.
L'
Let
is distinguished in
a relative p-base of
L'/K
L'
be an intermediate field of
L/K
if and only if there exists
which is relatively p-independent in
L/K. Proof.
Suppose there exists a relative p-base
which is relatively·p-independent in X
in
L'
finite subset E:
K(LP)(X').
X' U [x} Let
of
M' eM
X
L/K,
L'/K,
but is
then there exists a
such that
L'/K
If there exists a set
which is relatively p-independent in
not relatively p-independent' in
x
L/K.
M of
x t X'
and
be a finite set such that
6 X' U [x ] EK(L'P)(M'). such that
Then there exists a subset
K(L,P)(M') = K(L,P)(X',x,M")
relatively p-independent in
L'/K
and
M'
have the same number of elements, say K(LP) (M') = K(L P) (X' , x, Mil). [K(LP) (M') : K(LP)
J
= p
is impossible since
n
and
of
Thus
Hence
> p n- l > [K(LP) (X' , x , Mil)
K(LP)]
K(L P) (X', x,M").
K(LP) (M')
which
The converse q.e.d.
1.11.
Proposition.
mediate field of M
is
X' U [x} U Mil
is trivial.
that
M'
X' U [x ] U Mil and
n.
Mil
n M'
=
¢
L/K. and
Let Let
L' ,be a distinguished interM
and
M' C L'.
M'
be subsets of
L
such
Then any pair of the following
conditions implies the third condition. is a relative p-base of
(a)
M
(b)
M'
(c)
M U M'
Proof. such that
is a relative p-base of
such that exists
m e: M
a contradiction. e K(LP)(M'-m',M),
such that
is distinguished in L (LP) (M - m) I
L/K.
then
L/K.
If there exists m
Hence
If there exists
m'
M ee
M'
then by condition (a) there since
m c K(LP) (M I, M- m)
which contradicts condition (a).
K(LP)(M,M ') = L.
m
K(LP) (L') (M - m)
m't K(LP) (M' - m',M - m)
is relatively p-independent in whence
(c):
m e K(LP) (M',M - m),
m'
L'/K.
is a relative p-base of
(a) and (b) imply
L' (LP) (M - m},
L/L' .
L/K.
Finally
Thus
L
I
C
MUM'
K(LP)(M,M ')
L'
7 (b) and (c) imply (a): m s L'(LP)(M-m),
then
If there exists
m
M such that
=
K(L,P)(W)(LP)(M-m)
m
which contradicts condition (c).
Also
L
K(LP)(M',M-m)
K(LP)(M',M)
= L' (LP) (M) . (a) and (c) imply (b): p-independent in
L'/K.
exists an element p-independent in a
t
K(LP)(M'),
such that
a
but
Suppose
in
L'/K a
By condition (c),
L'
L'
L/K
K(LP)(M',M).
rn e K(LP)(M',a ,M-m)
is relatively
K(L,P)(M'). M' U fa}
such that
whence in
M'
Then there is relatively
by hypothesis.
Thus
Hence there exists
L'(LP)(M-m)
q.e.d.
Proposition.
Let
a purely inseparable extension minimal generating set. L'/K
L'
be an intermediate field of
L/K
such that
L'/K
L'
has a
Then every minimal generating set of
can be extended to a minimal generating set of
and only if
M
which contradicts
condi tion (a). 1.12.
m
is distinguished in
L/K
and
L/L'
L/K
if
has a
minimal generating set. Proof.
Suppose every minimal generating set
can be extended to a minimal generating set of M' U M. and
L/K,
Then
M'
and
p-base of
L/L'.
L/K,
of
Hence
L'
is distinguished in
By Proposition 1.11,
Since
L
= K(M',M),
L
M
L'/K
say
M' U M are relative p-bases of
respectively.
by Proposition 1.10.
M'
L'/K L/K
is a relative
= L'(M).
Thus
M
is
8 a minimal generating set of distinguished in M.
Then
M
L/K
L/L '.
and
L/L'
Conversely, suppose
L/L'.
Let
M'
minimal generating set whence a relative p-base of
Thus
MUM'
MUM'
L'/K.
Then
L/K.
L/K
since
L/K.
Proposition.
1.13.
be a
is a relative p-base of
must be a minimal generating set of
it's a generating set of
is
has a minimal generating set
is a relative p-base of
by Proposition 1.11,
L'
q.e.d.
Let
L'
be an intermediate field of
L/K. (a)
If
(b)
Let
L'
is separable, then
L/L'
is distinguished in
L/K.
L'
is distinguished in Proof.
LP
be algebraic and let
L/L'
(a)
Suppose
and let GUM m
L/L'
are linearly disjoint over
is p-independent in G
L.
K(LP)(M-m),
L'.
then
m
be perfect.
is separable algebraic.
L/L'
is separable. L'P.
L'
Then
(b)
K
Suppose
S.
L'/K
L,P(G) = K(L'P)
such that
If there exists
m
and
M such that
K(L,P)(LP,M-m) ELP(G,M-m).
L'
is distinguished in L/L'
L/K.
L'
Since
preserves p-independence.
denote the maximal separable intermediate field of L
L'
is
L.
perfect, this means
Suppose
and
Thus every p-base of
However this contradicts the fact that every p-base of p-independent in
If
M be a relative p-base of
Let
be a subset of
is a p-base of
then
L/K,
K
Then there exists
c
L,
c t S
Let
K
is S
L/L'. such that
9 cP
S.
(a).
Any p-base
Now
SP(B)
B
of
L,P(B)
L'
is p-independent in
L'
whence
and purely inseparable over cP
SP(B).
such that
SP(B).
S
S
is both separable
Thus
S
=
SP(B).
By the exchange property, there exists SP(B - b,cP)
b
independent in
L
LP(B - b).
by part
Hence
B
Hence b
B
is not p-
which contradicts the hypothesis that
L/L'
preserves p-independence.
Therefore
B. Extensions of type
A relative p-base is not necessarily
R.
a minimal generating set. when
L/K
1.14.
This difference can occur only
L/K
is said to be of type
only if every relative p-base of a minimal generating set) of
if
L
f L'
that
proposition.
(b)
L' (L P)
f
if and
is a generating set (hence
is of type
L/K
(a)
for every intermediate field
R
L'
if and only of
L/K
such
L. If
Proof.
p-base
L/K
R
L/K.
L/K
is of type
R,
for every intermediate field
field of
q.e.d.
has no exponent.
Definition.
1.15.
L = S.
L/K M of
(a)
If
L
=
such that L/K.
Thus
L'
L' (LP) L' L
then of
L/L'
is of type
R
L/K.
where
L'
is an intermediate
then
L'
contains a relative
f
L,
::l
L' 2. K(M) .
Conversely, if there
10
M of
exists a relative p-base L = L (L P),
L -:} L (LP)
(b)
L
::J
then
K(M),
L" -:} L
L"
of
is of type
R.
for every intermediate field
II
such that
such that
L' = K(M) .
where
1
L/K
L"
since
::J
K
and
L/K
L/L'
q.e.d. Lemma.
1.16. M
Let
L/K
be purely inseparable and let
{m ... } be a relative p-base of L/K. If there exist l,m2, positive integers e l,e ... such that e i < e i+ l, i = 1,2, ... , 2, ei-e l and such that }:=l is a minimal set over K, then =
{ml
L/K
is not of type Proof.
Suppose
verified that L/K r
whence
M' L
such that
Hence
M
r
p tmlm2
L/K
=
1
K(M').
m L = l r ei-e l i
1
By our hypothesis j
R.
is of type
ei+l-e i 1 1+
1=
1
Now, it is readily
R.
is a relative p-base of
Therefore there exists a positive integer e 2-e l
e
1, ... , r + l}
3-e 2
L . r
is a minimal set over
M
r
er+l-e r , ... Set K.
). Lr+ 1 = K(Mr ).
Clearly,
e er+l-e r 3-e 2 p p ' ... ,mrmr+ l }. is a minimal generating ,m2m 3 L /K. Thus L /K is minimally generated by r r r
e 2-e l
set of
elements while the intermediate field generated
by
r + 1
elements over
ble by Proposition 1.8.
Hence
L/K
L
is minimally r+ l K. However this is impossiis not of type
R.
q.e.d.
11
Proposition.
L17.
set
I L'
D(L) = [L'
Let
be purely inseparable and
L/K
is distinguished in
exponent if and only if every element of Proof. that of
Suppose every element of has unbounded exponent.
L/K
Then
L/K.
unbounded exponent, sequence
M
is of type
D(L)
E:
R.
and
R
M be a relative p-base
Let L
is of type
D(L)
Since
D(L).
has
L/K
is an infinite set and there exists a
mi e M, e i < e i + l, i = 1,2, ... , and is the exponent of m. over
[mil:=l
e.
where
because
L = K(M)
has an
L/K
L/Kl.
1
such that
1
K(mo) means K. The strict inequality can be achieved because L/K would have bounded exponent if L/K(m ... ,m has bounded l, i) exponent for any finite subset
[rJ?1
the sequence [m ,
1
set
j
}"'.' 1
of
n.
m.
J=
J
ei-e l 1=
1
1.
1=
and
1
of
i}
M.
Now consider
Suppose there exists a subsequence
l'
j = 1,2,...
e.1. ,
J
(with
J
nl = m 1
f .-f
P J
convenience and without loss of generality), then is a minimal set over whence
K(N')
K.
is of type
Let R.
N'
=
[n j}j=l'
However
R.
Thus the subsequence
[n j lj=l
fore the intermediate field
L' = K([ml
that every relative p-base of p-base of
L'/K
containing
[nj}j=l
m l
L'/K
ei-e l
[n.
J
Then
satisfying the conditions of Lemma 1.16 so that of type
If we
with the following properties:
f.
J
[ml, ... ,m
1
00
}. 1 J=
K(N')
D(L)
is a sequence K(N')
is not
cannot exist.
1:=1)
for
There-
has the property
is finite because a relative
can be chosen from the generating
12 ei-e l
set exist.
1
Thus if
and a subsequence of the above type does not M'
is a relative p-base of
exists a positive integer
t
t+l t K(L'P) = K(L'P ),
Hence over
K.
Since
L'/K
unbounded exponent. over
t(M')
K(L'P)
K(L'P)
t
D(L)
K(L'P)
with t
K(¢)
is of course immediate that if
L/K
1.18. set
S (L)
D(L)
=
[L'
IL
Let
t
L/K
Proposition.
L'
Suppose that
L
type
R,
:=
and
L'L"
has bounded exponent then q.e.d.
R.
be purely inseparable and
L"
Let
L/K}. S(L)
L/K is of
are intermediate fields of L"/K
has an exponent.
then both
L'/K
and
L"/K
The conclusion for
L"/K
If there exists a relative p-base K(M), L'
be purely inseparable.
L/K
and
Proof.
of
It
R. 1.19.
L'
has
a contradiction.
has an exponent if and only if every element of type
K.
K(L'P )/K
is an intermediate field of
I
C
is relatively perfect
K,
is of type
Corollary.
t
as its relative p-base.
=
K(L ,p )
M'P
is relatively perfect
has unbounded exponent,
Thus, by hypothesis,
every element of
such that t
so
But since
t
K,
=
L'/K, then there
such
L/K
If
L/K
are of type
is of
R.
has already been noted.
M of
L'/K
such that
then there exists a proper intermediate field
such that
L'
L*(L'P)
and
exponent, in fact, such a field is
L'/L* L*
=
L*
has unbounded
K(M).
Now
L"
=
K(N),
13 where
N
positive integer whence
L**
I-
t.
Set
else
L, Now
ed exponent.
L/L*
=
L
L**
L*(N).
L (N)
L'Lli
I
FC(L)
Proposition.
[L'
I
[L: L '] < 00,
L'/K
I-
that is,
L'
Proof.
(a)
so
Let
L'
L**(LP)
L/K
for a proper
I-
L
K(LP). R
q.e.d.
be purely inseparable.
Set
such that
L/K L/K}.
L/K
is of type
R
for every
L' e Fc(L)
such that
... , n
r}
L'/K
L of
E:
if and only if
R
if and only
Suppose
is of type
L/K
is of type
R
by Proposition 1.19.
is finite, is of type
then
L
K(M) (LP) L/K
and
K(LP) (M - m)
whence
LI
If
R.
L/L '
is
for some finite subset L' (n l' ... s n r) L. Hence L L'Lli, where L" K(n n l,.·., r),
L/L'
base of
is of type
L'/K
Fc(L).
suppose every proper intermediate field
LI
-c L* ,
=
L*L,P(N)
is cofinite in
is of type
L/K
for every
l,
t
would have bound-
L'/L*
L
for some
L.
finite, then [n
L**P
is an intermediate field of
is of type
(b)
R
L'
Suppose
(a)
L'
c K
which contradicts part (a) of proposition 1.15.
1.20.
if
=
t
Then
and thus
That is, L**,
NP
is a minimal generating set, so
and M
L
I- ¢
for some
If
R.
K(M),
because m
2 K(MP) (M - m) (LP )
E:
M.
L'
L/K where L
Conversely, of
lS
such that
not of type
M
K(LP). Now
L/K
R,
is a relative pLet
K(LP)
K(mP) (M - m)(L IP) .
2 K(MP)(LP ),
If
14 L' == K(mP)(M-m),
then
Thus
L'::::) K(mP) (M - rn)
type
R
Suppose
L::::) K(L P).
and so we have that
L/K
L' /K
is of type
q.e.d.
Rand
1
L
K.
is of type
R.
The converse is immediate q.e.d.
Definition.
extension
L/K
A subset
M of a purely inseparable
is called a subbase of
L/K
if and only if
MeL - K, L == K(M),
and for every finite subset
of
K] ==
is a tensor product over
Proposition.
1.22.
of
a p-base of b
over
L.
K}.
Set
B - K
(2)
K == KP (c) .
(3)
c
(4 )
L == K(B)
subset of
r
IT (K( m.)
i==l K
= 1, ... , r.
that is,
1
Let
L/K
C == (b P
i
be purely inseparable and b e B,
i
is the exponent
is a subbase of
and
L/K.
K.
i K c LP (c), i
1,2, ...
( 1) implies (2) :
Let
-
be the exponent of
K
r}
of the simple extensions
is a p-base of
B
: K],
(m1 , ... ,m
Then the following conditions are equivalent:
( 1)
Proof.
i
Then
L e Fc(L).
1.21.
B
is not of
Thus, by part (a), it follows that for every
Fc(L), L'/K
because
a contradiction.
which contradicts the hypothesis.
(b)
L'
L == L'(m) == K(M),
and let
Then, since
e.
1
KP(C)
(bI'" ., b r
.s K,
1 be a finite b.
1
over
K,
15 e
e
p l ... p r > [KP(C) (b e e p l ... p r
over
l,
... ,b
Therefore,
KP(C),
whence
K
B'
=
[b
such that
1b
e: B,bP
=
KP(B') Thus
C
n
Since C'
i
KP(B)
are linearly disjoint
K
Since
K == KP(C),
Then
c'}.
L == K(B),
KP(C). there exists a subset
is a p-base of
KP(C')(B') == K(B').
L/K(B')
[K(b p ... ,br) : K] ==
KP(B) == KP(C).
Thus
(2) implies (3): of
: KP(C)]
K and
K c L == LP(B) == KP(B).
C'
r)
K.
Let
(K(B'))P(B')
Hence
B'
=
is a p-base of
preserves p-independence, whence
L == K(B')
Proposition 1.13 and the pure inseparability of B == B'
whence
C ==
Since
i K == KP (c), i == 1,2, ... K
=
KP(C)
and thus
B
L/K.
by
Thus
c'.
(3) implies (4):
Now
K(B').
.
implies
C
is a p-base of
is a p-base of
K(B) .
i
Hence Thus
(K(B))P(B) == K(B) L/K(B)
p-independence, from which it follows that
(4) implies (1):
we have
K S LP (c), i == 1,2, ...
Therefore K S KP(B) .
K,
L
preserves
= K(B).
B U ... U B be any finite subl r set of B - K where every element of B. has exponent i 1. over K, i = 1, ••• ,r. Suppose that B. has S. o elements 1. 1. Since C consists of P e -th powers of elements of the p-base
(*) [L
p i+ 1
B
of
L,
pi (C) (B i+1, ...
[K(Bl'" .,Br)
K]
Let
it follows that i
) : LP
i+l
(C)J
=
s. 1 s P 1.+ ..• p r.
[K(B l, ... ,B r) : K(Hi, ...
J. [K(Bi, ...
Now :
16
K( B
p2 2 "'"
2 ].
pi+l pi+l K(B i+1,···,Br ), p
si+1
... p
sr
r-l
.
• ••
Since
): Kl.
i+1
P K( BI? 1+ 1 " ' " Br
we have
otherwise we contradict equation
sl sr s2 sr [K(Bl'" .,B r) : K] = P ... p p ... p B - K is a subbase of L/K. 1.23.
Proposition.
p-base of over
K
L. if
otherwise}.
Set b
Let [bP
C
L/K
i
b
p
(*).
Thus
sl 2s 2 rSr p ... p
Hence q.e.d.
be an extension and B,
i+1
i
B
a
is the exponent of
is purely inseparable over
K
and
=
i
b
0
Each statement in the following list implies the
succeeding one. (1)
L/K
is purely inseparable and has a subbase.
(2)
K
(3)
There exists a p-base
LP
and
1
are linearly disjoint, B
of
L
i
=
1,2, . . . .
such that
i
KeLP (C), i = 1,2, . . . .
(4)
If
L/K
exponent, then Proof. Then for all M: 1
is purely inseparable and of unbounded
L/K
is not of type
(1) implies (2): i
Let
R. M be a subbase of
1,2, ... , M ==
has exponent at most
i
where every element of
over
K
and every element of
has exponent greater than i over K. Since L = i i i i i i LP = KP (Mi P ,M.P1 ) := (LP n K) (Ml ), i = 1,2, ... let of
b.
J
over
K, j
=
1, ... ,r.
L/K.
Then
e.
J
1
M. 1
1
For any
denote the exponent
) ]=
17
P
el-i
i e -i ... p r > i:(LP
n K) (bi
i s ••• ,
i b Pr )
LP
i
n K]
i i e -i e -i 1 r Thus > lK(bi , ••• , b P ) : Kl == P ... p r i linearly disjoint over LP n K, i 1,2, ... (2) implies (3) : LP
over
K = (LP
n K,
n K)(C o)
and
Co
j
n K, LP
i-I
in
Co
K
C ., j == 0, ... , i-I,
n K
(LP
i
i-I
n K)(
and
K
are
such that
is p-independent in
J
i
are linearly disjoint
K
and
there exists a set
that there exist sets C. c Lp
LP
Since
LP
L.
Suppose
such that i-2
' ... , Ci-I ) and i-I i-I i-2 P P is in L Then cP c p-independent U U U C. 1 0 1 1i i i-I P cP c U C.P 1 c LP n K and is p-independent in U U 0 1 1i i+l i+l Since LP and K are linearly disjoint over LP LP n K, i+l i i+l and LP n K are linearly disjoint over LP LP n K. J
ee
, ci
...
.
i Thus there exists C. c LP n K such that 1 i i+l i i-I LP n K = (LP n ,ci ' ... ,C i) and i
U
U Ci
LP
is p-independent in
i
Hence there exist i i sets C. , i == 0,1, •.. , C. c LP n K, LP n K == such that 1 1 i i-I i i+l i-I n and c P U c P , ... ,C ,C Pl U C. (L P U 1. i) 0 1
...
is p-independent in K == (LP
i
L
p
i
,
l'
n K)(C o , ... ,C.1- 1)
== 0 , 1 ,
...
whence
K
-i
00
i
1,2, . . . .
Furthermore,
U
i==O
1
Thus
is p-independent in
L.
18 -i
co
U
Augment C
=
[bP
to a p-base
i=O i
1i
B,
i
=
(LP
i
K
U C ..
C*
i=O
bounded exponent. K(LP)(B),
that Set
M c B. M
=
i
0
=
i
K
if
b
is
otherwise},
=
=
n K,
C
1,2, . . . .
Let L/K
B
be a p-base of
L
Suppose first that
Hence
satisfying
is purely inseparable and of unL
K(B).
there exists a relative p-base
B - K.
exponent
over
1,2, ... , C*
K(M) S K(B)
L.
C
M
e.1. (e.1. < e.1.+ 1)
over
K'
M
Then since of
L/K
Suppose that
Then by Proposition 1.22,
Hence there exist
L/K.
b
1.
condition (3), where
=
Then for
L.
KeLP (c), i
Thus
(3) implies (4):
L
and
n K)(C*), i
co
since
of
is the exponent of
purely inseparable over K
B
1.
M
=
K(B).
is a subbase of
such that
K(M'),
L
such
m.
1.
has
where
[m l,m ... J, i 1,2, . . . . Now it is readily verified 2, [m l,m2, ... J is a relative p-base of L/K' satisfying the
M' = M that
hypotheses of Lemma 1.16. Mit
of
L/K'
M' U M" Thus
such that
Hence there exists a relative p-base L
K'(M
It
is a relative p-base of
L/K
is not of type
R.
1.24.
Proposition.
Let
without exponent.
Let
)
By Proposition 1.11
.
L/K
and we have
K(M',M
It
q.e.d. L/K
be purely inseparable and
M be a relative p-base of
L/K.
Each
statement in the following list implies the succeeding one.
)
.
19
(1)
M
(2)
n K(M-m) meM
(3)
n K(M- m)
is a subbase of
meM
(4 )
=::
K.
.f
LP
i
L/K.
,
i =
is not of type
L/K
R.
(1) implies (2):
Proof.
1,2, ...
This is immediate from Definition
1.21.
(2) implies (3): integer
e,
(3) R.
e
{e
m
c
I
m
L
over M}
E:
If
for some positive
which is impossible. then
L => K(M) ,
For all K(M- m).
m e M,
[e
l,e 2
, ... }
m.
1
[m.}':' 1 1 1=
s
let
e
tern 1m
e,
E:
M}
whence Thus there
1
ei-e l mlf Now
t
K(M - mJ .
ml?1
ei-e l
Thus
is a minimal set over
K'.
That is,
satisfying the conditions in Lemma Hence, as in the proof of is not of type
1,2, ... ,
e., i =
implies
L/K
is unbounded
such that
corresponding to
M' == M -
denote the
m
tern 1m e M}
Then
is such that
is not of type
L/K
contradicting condition (3).
For the sequence
e
has a least upper bound
n K(M-m), meM
exists a subset
=> LP
meM K
C
L = K(M).
exponent of
LP
e
implies (4) :
Suppose
else
LP
then
n K(M- m)
If
R.
(3)
, ... } {m l,m2
1.6 (with
implies
(4)
K
I:
Set
K(M - mi ) ei-e l }:=l
[mlf
is a set
replaced by
of Proposition
K').
1.23, q.e.d.
20 1.25. If
Proposition.
L = K(m l,m2, ... )
Let
L/K
be purely inseparable.
mi e K(mi+l)J i
where
= 1,2, .•. ,
then
j. 1
the intermediate fields of 0 < j. < e. 1
(e
1
i = 1,2, .. 'J
i
Let
K(mo)
K, K(ml
m i
means
over
K(m
are whence
fields of Let
K'
then
K(mt) =
are
).
since
),
is finitely generated.
0
i=l
union over
for all
1
K(ml
C
c
)
::::J
of
K(c)
K(mi -1) C
bounded function of Example.
[K'
for some
by the previous argument.
1.26.
1
K(ml
1
If
c
c.
o
J
L'
L'
and
L'
and
Hence
has exponent •
Thus whence
for some integer ms+ l
L'.
By the division algorithm,
and
K(L P),
L'
By Proposition 1.25,
K(mi)
pt-s_l
Since
L'
by showing that for
remains irreducible over
Since
appear
Hence the irreducible polynomial of
over
L'
L.
s > O.
are linearly disjoint over
K'
L'
L'
for some integer
every proper intermediate field K'
K(LP)/K
Now suppose there exists an intermediate field
such that
L' n K(L P)
Hence by
n
LP.
c ' e L'. J
over
=
L) . J
and J
23 Writing
x
-p t-s-l
t-s-l
x
s-t-L
=
k-Pk 0
linear disjointness of
I
=
j
t-s-l
k 1 -- y P
an d
t
( mPt ) P
(* ),
we get for the example
in terms of
P
Thus
K(LP)
and
J.
t
J
J
By equa t '10n
x s+ l - y. r.
0,
0, ... ,pt-s - 1)
x s+l ..'" L'
t-s-l
+ k- P
1
L'
x -p t
Hence, by the
over
and since
is linearly independent over
-1
a contradiction.
x s +1
For the example in part (b), we get 1 t-s-l 1 - x s+2Y= - kl for e K. By an argument similar to that of the
1
=
suitable
ko,k l
example in part (a), we obtain If
L/K
is of type
yP
and
R
R.
above and
For instance, if
=
F
L(YP
-1
),
relatively p-base of
L/K
e L,
a contradiction.
is a finite degree purely
F/L
inseparable field extension, then type
-1
is not necessarily of
F/K
is the example in part (a) is a
then
F/K
such that
F:J
K(M).
However the
following proposition gives a criterion for an extension to be of type 1.27. extension
R
when
L/K
Proposition. L/K.
If
some positive integer
L/K e,
is of type
F/L
R.
be an extension of an e is of type R and L = K(PP ) for
Let
then
F/L
F/K
is of type
R.
24 Proof.
Let
e K(LP)(NP )
L
=
N be a relative p-base of e+l e e K(F P )(NP) = K(F P ) = L.
tains a relative p-base e
K(F P )(N)
=
K(L P )/K L/K
=
is of type
is of type
=
K(M)(N)
Corollary.
1.28. e
L(N)
M of
If R
L.
Thus
F
F/K.
Then e Hence NP
= q.e.d.
K(N).
L/K
con-
is an extension such that
for some positive integer
e,
then
R.
C. Special generating systems.
We now derive and analyze some
generating systems associated with the towers -1 -2 and K C KP n L c KP n L c Proposition.
1.29.
Let
be subsets of
K(L P )/K, i
The subsets
Bi
be purely inseparable and let
such that
L
i
a relative p-base of A.
L/K
=
M.
1
-
M.
1+
1
and
Mr:'
i
1
is
0,1, . . . .
= Mi_ l - Mi, i =
1,2, .•. ,
satisfy:
00
(a)
U B.
i=l
i
Br:'
(2)
For all
1
i+l
00
U BP.
j=i J
i
and for
c K(L P
(1) K(L P
(b)
MO
1
i+l
1,2, ... ,
i
i i )(Bl+ 1,Bl+2 " " ) '
b e: B. , 1
b
has exponent
i
over
)(B i - b,B i+ 1 , ... ). is p-independent in
i
K(L P), i
=
1,2, . . . .
25 (c)
The cardinalities of
B.
1
and
M.1- l' i ; 1,2, ... ,
are invariants of the extension. B. B i
Mo of L/K(LP ) and subsets such that the conditions in (a) are
Given a generating set
in
L, i := 1,2, ... , co
satisfied, then K(L P
p-base of
M.
1
i
U B. j:=i+l J
==
)/K,
i
=
ME?
is such that
i
is a relative
1
0,1, . . . .
p i-l is a We first prove (b) . Since M.1- 1 i-l relative p-base of K(L P ) /K, we have for all b c B.1 that i-l i i-l i-l i-l bP ¢ K(LP )(Bl - bP ,Ml ). Hence i i i+l iii is p-independent of b P ¢ KP(L P )(Bl - b P ,Ml)' Thus BE?1 i i i i is p-independent in K(L P ). ThereME? in K(L P). Also ME? Proof.
A.
1
1
co
U BE? j=i J
for
i
is p-independent in co
For every
To prove (a), we first show that m
Mo'
Hence
there exists a positive integer m
M. 1
co
M
o
U
ee
i=l
B .•
M1? 1
i
n K
p.
such that co
Thus
n M.
i=O
P
;
1
mP
i
K.
whence
Now, to prove ( 1), we observe that
1
i
C
since
i
K(L P )
K(L P
i+l
i )(Ml)'
(1) implies that for all (b) implies that
i
Ml_
l
b
To show (2), we first observe that i i+l B b P e K(L P )(B i - b,M i) and i,
is p-independent in
i
K(L P).
Therefore,
26 if we set
S =0 (M U (B - b)) U [bPi}, we have that S is i i i i+l p-independent in K(L P )(S) K(L P )(B i - b,M i) by Proposition i i+2 )(Bl - bP,Ml), whence 1. 7. Consequently, b P t KP(LP i+l K(L P )(B i - b,M i ) · To prove (c), consider any other chain say N =0 No
Nl
...
base of
K(L P )/K, i =0 0,1, ...
i
= 1,2, ...
i
Ni
Fix
i
by (b), there exists a subset i
J..
is a relative P-
- N C =0 N i_ l i i,
Set
i
and set
i i K* =0 K(K*P)(Ml ) =0 K(K*P)(Nl ) i i base of K*/K and U J.. 1-
i
such that
:::J ••• ,
.
K* =0 K(LP). Then i Since is a relative PJ.. i M.PJ..- 1 is p-independent in K*
Go
of
K
such that i
i
i
Now ) U Bl is a p-base of K*. J.. J.. i-I i i i-I i-I = (K*(Ml_ I ))P =0 (K*(LP ))P =0 (K*(Nl_ I ))p = K*P(Nl 'Cl ). i i Therefore Go U J.. U is also a p-base of K*. Since :I.. i i and are relative p-bases of K*/K, there exists a :I.. :I.. i as well as subset G of K such that G0 c G and G U MI? Go U Ml
G U
NP.:I..
:I..
i
are p-bases of i
i G
o
U
:I..
i
G - Go' of
U :I..
and
K*/K*P(G 0'
G i
:I..
)
We already know that i
and
:I..
K*. :I..
-
Go'
and
i
:I..
U
i :I..
are p-bases of
Now
are relative p-bases of
K*/K.
Hence
are each minimal generating sets i
K*/K*P(Go,Nl)'
IB.I =0 IG - G01=0 Ic.l. :I.. :1..
K*.
1M.J.. I =0 IN.:I..I
respectively. because
Thus and
NP.:I..
i
27
B. Bl?
i C
J
Hence
For
P
j
i-j
K
(L
P
pj < i, B.
i+l
Thus
J
i
i
i i+l i K(L P ) = K(L P )(Ml)' Therefore it remains to be shown i i Ml? is relatively p-independent in K(L P )/K. Suppose
that i Ml? is not relatively p-independent. l i i i i+l P such that b P e K(LP )(Ml - b )
.
Then there exists Let
k
b
M. l
be the smallest
positive integer for which If
k
=
1,
Bi+k contains such an element b. then (2) is contradicted. Suppose k > 1 and that
cannot be represented as an element of without using elements from
U
pi Bi+k- l.
Then an element
i
bP . i
However this i+l iii ) (Bl+ k - b P , Ml+ k) contradicts the minimality of k. Thus b P e K(L P i+k-l i+k i+k-l i+k-1 i+k-1 from which it follows that b P e K(L P ) - bP ) (Bl+k , Ml+ k from the latter set can be exchanged for
P
i+k+l
=K(L
i+k i+k-l i+k-1 i+k-l P , Ml+ k ,Ml+k ) (Bl+k ) - b
But this contradicts (2).
Hence
i
K(L P )/K. 1.30.
Ml?
.i.
i
K(L
P
i+k+l ) (B i+k - b,M i+k)·
is a relative p-base of q.e.d.
Definition.
If
L/K
is purely inseparable and
Bl,B •.. are subsets of L satisfying the conditions in (a) of 2, Proposition 1.29, then {B ... } is called a canonical system l,B2, of L/K and Mo is called canonically ordered.
28 1.31.
Corollary.
A.
If
L/K
if there exist subsets
M. 1.
in
L
Ml
i
such that
is a minimal generating set of
then the subsets defined by
is purely inseparable and and
M1.' -::> M.1.+1
i K(LP )/K, i = 0,1, .•• ,
1,2, ... ,
B.1. = M.1.- 1 - M., 1. i
satisfy: 00
(a)
U B. = M
and for i = 1,2, ... , a i i i 1. -c 1.+ 1.+2 " " ) '
i=l 1.
( 1) (2)
For all
b
B., 1.
b
has exponent
i
over
K(B i - b,B i+ l,···)· Conditions (b) and (c) of
B. in
1.29 hold.
Given a generating set
L, i
1,2, ... ,
M
o
of
M. 1.
erating set of Proof. i
then B,
1.
If
U
B.
is such tha t
j=i+l J i K(LP )/K, i=O,l, . . . . i
i 1.
1.
i
i K(LP )/K.
Bl,B
2,
...
1.
is a minimal generating set of
Definition. are subsets of
If
L/K L
i K(LP )/K,
For (1) ,
1. pi pi _ pi+l pi pi then K(B. 1,B. 2"") - K(L )(B. 1,B. 1.+ 1.+ 1.+ 1.+2"") i+l P - b,B i+ l,···) = K(L )(B i - b,B i+ l,···)· 1.32.
B.
is a minimal gen-
is a minimal generating set of
is a relative p-base of
note that if
and subsets
such that the conditions in (a) are satis00
fied, then
L/K
(2) and i
K(LP )/K,
and q.e.d.
is purely inseparable and
satisfying the conditions in (a) of
29 1.31, then
Corollary
erating system of and
L/K
[B
L/K,
l,B 2
, ... }
is called a canonical gen-
Mo is said to be canonically ordered
is said to be canonically generated.
The elements of
00
U B.
i=l
are called canonical generators of
1.33. of
L/K
Example.
The existence of a minimal generating set
does not insure the existence of a canonical generating
system of
L/K:
Let
P
and
1
L -1
zP
However
=
K( xi
i
I
i
1
=
K(LP)
1
'
, ... ) (zp
P.
Let
K
12 , zP , ... ) .
=
2
K(L P )
=
... ,
i
K(LP )/K
whence
. )
Then
is a minimal generating set of
1,2, ... }
=
z,x l,x , 2 p(z,x l,x , 2
be a perfect field and
independent indeterminates over
[xI;>
L/K.
1
L/K.
can have no
minimal generating set.
1.34.
Proposition.
Suppose
L/K
and has a minimal generating set subsets
M.
in
1
M
such that
0
i
K(LP )/K
exponent over Proof. and
il P M. 1- 1
existence of
M.
1
There always exist
M
o
M.
1+
1
and
MI;>
i
1
is a
1
relative pbase of set of
=
M
is purely inseparable
Then if and only if
MI;>
is a minimal generating
1
B.
1
=
M.
1
1 -
M.
1
is of bounded
i
K(MI;>1 ), i = 1,2, ... Suppose
B.
1
is of bounded exponent over
is a minimal generating set of being assumed).
Then
i
K(Ml) (the and
30 i
since Bl? 1.
i
Assume whence
is a relative p-base of K(L P )/K we have i+l i i+l i i+l i P C K(L )(Ml) K(Ml_ l )(Ml ) K(Bl )(Ml)' i i i .i-i-n-rL i+n i+l Bl? C K(Bl? ), Bl? c K(Bl? Then l. ),n )(Ml 1. 1. 1. 1. i i i i+n+l i Bl?1. E K(Bl Thus ) )(Ml ). Hence Bl? K(Ml 1.
.
i
i
K(L P ) induction.
Therefore the desired result follows by
1.35.
Corollary.
relative p-base of of
q.e.d.
The converse is immediate. If
L/K
L/K
has an exponent, then every
yields a canonical generating system
L/K. When a subset
mean that
A
A
of
L
is called a subbase over
is a subbase of
K(A)/K.
K,
we
A subbase is called
equi-exponential if every element in it has the same exponent. If
L/K
has an exponent
e,
1.31 and
then Corollaries
1.35 and the following Proposition show that since Be+ l
= Be+ 2 = ...
manner: L
If
B.1.
f ¢,
¢, Bl, ... ,B e then
B.1.
can be chosen in the following is taken as a maximal subset of
with respect to the property that
subbase over exponent of
1.36.
K(B.1.+l' ... ,B) e
Suppose
The conditions in (a) of Corollary
L/K
has an exponent
e.
1.31 are equivalent to the then for all
If
has exponent
K(B;... - b,B.1...-'1' ... ,
over
is the
i
1, ... ,e.
following conditions: i
and
i
with exponent
L/K(B.1.+1'" .,Be ), i Proposition.
is an equi-exponential
B. 1.
)
and
e B., 1.
b
i
L
is the
31 exponent of Proof. hold.
LP
i
B.
over
Suppose the conditions in (a) of Corollary 1.31
If
then clearly for all
E K(B
over
K(B. - b,B.l+ l' •.. ,B e ).
b,B
i
b,B i + 1 ,
K(B i
K(B i - b,B
,B
i+ l,
i+ l,
... ,B
,B
L
i
c
n
be:B. l
must have exponent
since
e)
b
has exponent
Then (2) is immediate.
K(B. - b,B·+ 1 , ••• ,B ) == K(B';+l' ... ,Be). e
Assume
Thus
If we show that this implies
== 6k
0 < t
and t
s.
t t l s b l ... b s s
t l· .. l,
.. "
Since
i == 1, ... ,e.
< P L
i
where
k
By
q t 1'"
(2),
k
t l··· t s
q
t
t l· .. s
bP
i
e: K(M.), k J
j-1
= (bP)q
i
Bl?l -c
t 1 · .. t s
e: K'.
,
e: K', b 1,···,b e B. s J
t
s
over
< j
K(M.), J
q q pj pj e: K (M.)( Bl , .•• , B.) J
J
the products
implies
t t (bi) l ... (bi) s
== 0
for
t
l
+ ... + t
form and
Since q
K'
j-i
Thus
a linear basis of k
q == p
b e B.l ,
Now for
has exponent
Thus
over
and
then ( 1) holds by induction. i
i
Now
Let
bP
i
Conversely suppose the conditions of
e).
the Proposition hold. P
L
and
e)
b e: B., .i,
s
> O.
Hence
q.e.d.
.
32
If empty
of Proposition 1.29 or Corollary 1.31 are
B. 's 1
deleted and the remaining ones are relabeled keeping them in the same order, then we get similar results by making the substitutions
pe
nonempty.
i
for
pi
and
Bi
for
Be.'
where
B., {B.} ].].
and and
is
].
In this case we have
the cardinalities of
Be.
1
M.1- l'
i
e.
and
1
1,2, ... ,
are
invariants of the extension. 1.37.
Proposition.
Let
L/K
be purely inseparable and
let
M be a generating set of L/K. Suppose there exist non o empty disjoint subsets B.]. of M such that for distinct o positive integers
(a)
e
i
i,
1,2, ... :
co
U B. =
Mo i=l ]. q. 1 B. (1) c ]. (2)
and for q.
For all
b
=
i
1,2, ... ,
q.
B
E:
where b
i,
=
has exponent
Then there exists a reordering of the as in (1) and (2) such that (1),
qi
B.].
P
e·1
e..i,
, over
e.].
with the same
(2) and also (3)
el < e
2
1
such that
Jt
that
Then
B.
Ji
K =
¢
and
B. U B. U ... c K Ji+s Ji+s+l r
K (B. r
Ji+l
, ... ,B.
Ji+s-l
)(B.
Ji
b)
Then
since
Thus there exist finite sets
(t = i + 1, ... , i + s - 1) and B'.' c B.J. J. Jt q. /p J. b , ••• , B '.' ) (B ': ) , where e K (B '! r Ji+l Ji+s-l Ji B.
n
contains all but a finite number of the B. 's.
K (B. , ... , B. ) ::2 K(M:). r Ji+l Ji+s-l B':
j ..
r
-
such
rb}
Jt
¢
if
.U B.
B. c K. J r t K
r
Since
satisfies (1) and there exist finite sets
),
J i+s-1
J. /»
q. i
K r
B:
Ji+l
+ 1, ... , i + s - 1)
, ...
Ji+s-l
such that
Jt
b
and
) (B'.' ), Ji
q.
J
Jt
t
K (B r
=¢
if
J t+1
, ••• , B :
Ji+s-l
c K.
B.
1 + l, ... ,i + s -
1),
where
However this contradicts the above degree
r
Jt
(t
)
relation.
q.e.d.
L/K
In the above we deduced invariants for intermediate fields
i K(LP),
intermediate fields
(KP
-j
i = 1,2, . . . .
n
i L)(LP ) ,
by use of the
By considering the
we shall derive
additional invariants associated with a subclass of canonical generating systems.
But first we derive invariants for
by use of the intermediate fields Set
=
KP
-j
n
KP
-j
n L,
j
L/K
= 1,2, . . . .
= 0,1, .•.
and for the composite of j-i j-l with L:t:> set L .. =L·IL:t:> , j >i 1,2, . • . . J J For each positive integer j, let T.. be a relative p-base of JJ L·/L . . l' Making use of L:t:> 1 c L. 2' we find that if j > 1,
J
then
L.
J
j
J,J+
L. l '
J- ,J
J-
L.
J-
we can select from . l' J-,J·/L.J- 1 ,J+
L. 1
L,
-
L:t:> . J = L.J- 2 J,J+
T:t:>.
JJ
J-
JJ
= L. 1
a relative p-base
Now, with
relative p-bases as follows:
j
. J- ,J+
T. 1 .
J- ,J
JJ
Hence
of
fixed, we continue to select Let
k < j
and suppose we have a
35 T . kJ
relative p-base
/TL -k,j+l
h
of
Then using the fact that
Hence we can select from
that is contained in we have
Lk-1
of
a relative p-base
For each
j,
this process ends after a
finite number of steps, namely, with the construction of
T l j.
We can think of this process as filling boxes in an upper triangular infinite grid.
¢
Tk j
for
When
L/K
has an exponent
e,
then
k > e.
j
co
U T
Now define of
j+l
L _ , k l
j:=k k,. J
is a relative p-base
Since
L k , k- L :2 ...
and
it follows that
T k
j+1 2
is relatively p-independent co
For convenience, we set
in i-j
T!?1J
(j -> i
=
1,2, ... ,
j
=
structed for
L/K
L/K
so
N.. n N.. 11 JJ
and
T .. JJ
=¢
=
N .. JJ
when
UT ..
and
j=l J J
::>
N.J- 1 , J.
i
.f
N ..
=
U T..
JJ
j.
The collection of subsets
and is denoted by
[T .. }. 1J
is called a lower tower set of
From the definition of IT .. 1 1J
and
INI
==
::>
The subset
00
N
1J
T.. con1J in the above fashion is called a lower tower
Definition.
1.38.
system of
1,2, ... ),
N:=
T.. and N 1J are invariants of L/K.
L/K. it follows that
36 Proposition.
1.39. L/K.
Then
1 < i < j
IN . . . l+J,J
= 1,2, ..•
.
[T .. }
Let
N.1)·1
be a lower tower system of
1)
is an invariant of
L/K,
= L.1., J+ . 1 ( T. .) a s a c ompo sit e of 1J 1.J Ll? . and L. . lover Ll+l,j+l' Let A be a relative p1.+1,J 1.,J+ base of L. . l/Ll? 1 l' Then there is a set A* c A such 1., J+ 1.+, Proof.
that p
A*
Consider
L ..
1 .. 1. J·/L!? 1.+, J
is a relative p-base of
P
Since
L.
(A* U T .. ) U (A - A*) and 1.J (A* U T .. ) U (rrJ? 1 . - T .. ) are relative p-bases of 1.J 1+ , J 1.J L 1.J .. ILl? . l' Since A - A* is independent of the lower tower 1+ 1 ,J+ L 1J .. c L.1.+1 ,J+ . l'
both
system and the cardinality of a relative p-base is unique, we deduce that
every choice of
1,(x) = max j k
I xP
denoted by
1, . s
k
term in the chain
1.41.
L/K.
x k
T
IN.+ 1 · 1.,)
N .. )J
then its length
j j,
c T. k .}. )- ,J
for
. ..
N.. 1 1.)
1S
T .. , J)
))
)
1, . , N. 1, . J )- j')
::J
The length of
t(x)
1,. = maxp(x) Ix e T .. }.
is defined by
By the definition of
with length
IA - A*I
q.e.d.
defined by
invariant
=
1.)
If
)
J
[T .. }.
Definition.
1.40.
.1
IN. 1 . - N.. \ = \ T l ? l ' 1.+ ,J 1+ ,J 1J
N. 1, . )- j')
::J
is the last non-empty ::J
...
::J
Nl j·
Also, the
is the number of elements in
T ..
))
j - i - 1. Definition.
The numbers
Let
INI, IN i j
lower tower invariants of
!,
[T .. } 1.J ]Ni+l,j
L/K.
be a lower tower system of
-
N.. 1, 1.)
1,. )
are called the
37
1.42.
Proposition.
lower tower systems of ing
and
g
of
N
onto
[T .. }
If
geN .. ) == N ., 1 < i < j
(2)
t(x)
INi+l,j
-
Hence when
j
i < j
be extended to a
1 -
N.
gj
onto
from
onto
Nj j "
I
any
Starting with
"
1
1 .,
J-"'j' J
i == j -
preserves lengths.
1
,J.
-
1
.I
-
J
such that
J
gj
generating set of
1.44.
I
L/K
Proposition.
Ni j
i < j
.
onto
onto 1 -
can
Ni+l,j' 1
maps
mapping Ni j
g
1 -
1
is now defined
g. 's.
q.e.d.
J
A lower tower system
and
L/K N
If
[T .. }
is called
if and only if
is called a lower tower
if and only if (a)
1.
t j, ... ,j
This mapping automatically
The required mapping
), j == 1,2, ... ,
J-
N..
for
-
i == j
mapping of
a lower tower generating system of L. == L. l(
for
N. 1 , J. we can construct a
t., ... ,j.
Definition.
IN .I
IN .. 1 ==
mapping of
Nj j
to be the union of the
1.43.
x c N.
and note that
N.. 1 ==
and
1,2, ...
for every
== t(g(x))
Fix
mapp-
1- 1
such that
(1 )
Proof.
are any two
then there exists a
L/K,
N'
[T . }
and
L/K
L == K(N). has bounded exponent,
then every lower tower system (set) is a lower tower generating system (set). (b)
If
L/K
has bounded exponent, then the set of e x, monomials of the following type: TI x a ex < p t(x)+l ,
xeN
38 where all but a finite number of the basis of
are zero, is a linear
x
L/K.
Proof. L./L. 1 J
e
(a)
for every
J-
(b)
T.
In this case,
is a relative p-base of
J
j. e
o
x x
II
By part (a), the monomials
< e
XE:T.
x
where
< p,
J
all but a finite number of the of Le
L./L. 1. J
are zero, form a linear basis
x
Since we have the sequence of fields
J-
...
e
La,
L/K
has a linear basis consisting of the basis
elements of the intermediate extensions. exponents, For
L/K
-
h(x) = j
has the given linear basis.
T .. , JJ
x
Thus, combining
the height of
q.e.d.
is defined to be the integer
x
l. 00
Let
U T .. j=l JJ
denote a lower tower set
N
of
T .. is a relative p-base of L . / L. 1 (Ll? 1)' T .. J JJ+ JJ JJ P) P), relative p-base of L (L / L l (L j = 1,2, ... j j_ contains a relative p-base of L/K.
Ms·
Example.
not be a minimal generating set of L = K(ZP
-3
,zP
-2 -3 P -1 x + yP ),
fect field and Then
{T .. } 1.J
T = {zp 33 T22
T12
-3
x,y,z
-3
xP
-1
.
Thus
where
K
L/K
=
p(x,y,z),
P
¢, TIl = {yP
-2
-1
}
},
T = {zp 23
.
Hence
-2
}, T
-3
,zP
P.
where:
L/K
{zp
need
is a per-
are independent indeterminates over
+ yP
N
Let
L/K:
is a lower tower system of ,zP
contains a
lower tower generating set of
A
Since
L/K.
13 -3
= {zp xP
-1
-1
},
+ yP
-2
,yP
-1
}
39 is a lower tower generating set of zP
-3 P -1 -2 x + yP }
1.46.
L/K
while
{zp
is a minimal generating set of
Proposition.
-3
,
L/K.
The following conditions are equi-
valent. (1)
Every lower tower set
p-base of
(3)
Lj
T..
implies
(2)
immediately.
is a relative p-base of
JJ TIl
j
=
Then
T ..
JJ
If
1, ... , i.
p-dependent set in
has exponent
j
=
J-
J+
1,
,
L/K
1,2,... .
i+l
U
j=l
N
T
jj
Li+l(LP)/Li(LP).
=
holds.
T ..
JJ
1,2, • . . .
Let it hold for
Lj(LP)/Lj_l(LP)
is a relatively
Since
j = i + 1.
N
is a relative p-base of Thus
1)'
= 1.
which contradicts (2).
(3) holds for
(2)
is not a relative p-base of
Suppose (3) holds and let
Then
which
Ll(LP)/K(LP).
is a relative p-base of T. 1 . 1
is a relative p-base of
L/K.
J
(3) holds if j
then we have that
induction.
L./L.
Lj(LP)/Lj_l(LP), j
has exponent 1, .i ,
L/K
Suppose
must be a relative p-base of
Since
= 1, ... ,
of
are linearly disjoint over
contains a relative p-base of
for
N
= 1,2, . . . .
(1)
Proof.
(2),
is a relative
L/K.
Lj_l(LP )
and j
j
L/K
There exists a lower tower set
is a relative p-base of
By
of
L/K.
(2)
Since
N
Thus
T.
1 . 1
2)
L.
Hence (3) holds by
be any tower set of Lj(LP)/Lj_l(LP),
is relatively p-independent in
L/K.
40
Clearly
L = K(LP)(N).
K C KP
We have refined the chain of fields KP
C
-i
n
L
... ,
i = 0,1, ... ,
n LP
(1) KP
-i+l
L
n
KP
and
L
K(LP)
-i
n
j
-1
n
L c
by introducing the fields
1,2, ... ,
), j
between
We now refine the chain of fields
L.
i
K(LP ) 2 ... ,
..•
q.e.d.
Therefore (1) holds.
i
=
0,1, ... ,
by introducing the
1,2, ... ,
between
fields i
K(LP )(KP
(2) and
K(LP
i-I
1.47.
-j
n
LP
i-I
),
j
).
Definition.
The ascending chain of fields in
is called the lower tower of
L/K
(1)
and the descending chain of
fields in (2) is called the upper tower of
L/K.
We now give a construction of the relative p-bases of the upper tower of fields.
Let
be a relative p-base of 00
=
L/K(LP)
relative p-base of a relative p-base of 1,2, ... , M 21
2 K(LP)/K(LP )
=
¢.
and
K(LP
2
and
U MI'
Then
1,2, ...
L/K.
2
Now
j=l
Ml'
J
J
is a
clearly contains say
M2 j,
j
=
00
Then
U M2 ·
j=2
J
Suppose
is a relative p-base of i-I P M. 1- I
00
U M .. . . 1J J=l
is a
i-I i is a relative relative p-base of K(LP )/K(LP) where M.. 1J i i-I i i-I P p-base of K(LP )(L; contains a )/K(L Now j
Ml
41
relative p-base of
say
¢.
j = i,i+1, ..• ,M.l+ 1 ,l. p-base of
i
i+l
K(LP )/K(LP
M:!?l
Then
)
i
= i
l+ 1 ,J.,
oo
U M. 1 . . '+1 l+ ,J J=l
K(LP )/K.
and
M.
is a relativE
The construction of
the
M.. 's can be thought of as filling boxes in an upper lJ triangular grid. When
L/K
is purely inseparable we have that
is a canonical system of -i+1
...
kt, .••
is linearly independent over
Thus,
and
xl' ... , x r E X
S = {K.
J
I K.
is modular} .
containment.
Now
L E S
J
is an intermediate field of
Then
S
whence
L/K
is partially ordered under set S =I
¢.
Let
S·
be any simply
ordered subset of
S.
K* =
Let
Gc ES ,Kj J
but arbitrary positive integer. LP
over k
t
E K*
and
x
t
independent over hypothesis that LP
over K.
i
n
c
i
Let
EX, t LP
i
=
l, ... ,r.
n K.J
over
J
span of
(xl""'xm}
K. E S'. J
J
E S'.
Jo
K.
J
over
E S'. LP
i
0
is linearly Make the induction
is linearly independent
Let
If
n K.J
is a linear combination\of all
E S'.
K.
m < r,
K.
xl
=
S'
a
I
(K.
J
J
is in the linear
x m+ l for all
E S'0'
K.
J
LP
over
xl' •.. , x m
i
Pi
=L
n K.) J
n K*.
Hence, there exists
K.
Jl
E S' C S' 0
is linearly independent over fore, by induction, there exists
K. J
E S'
is linearly independent over
kl, ...
n
K.
J
for
n K*.
However, this contradicts the linear independence of
[Xl' ... , x r }
then
Equating these linear combinations, we find that
0
the coefficients all lie in
over
E S',
K.
is clearly linearly independent for all
K.
be a linear basis
Clearly
for any
Then
J -
i
where
for some
Jo
LP
C
n K*.
[xl' ... ,xm}, I
K.
X
be a fixed
i
Let
.
C
independent over minimal element.
Kj , kl K*.
= ... =
Since
Therefore,
=
O.
K* E S
n
K..
Jl
such that
remains linearly independent over
E K*
i
Thus,
K. J
(xl"
•. , x } r
Since
J
whence
There-
E S,
K .•
X
r}
such that
-
LP
(xl' ... ,x
is linearly S
has a q.e.d.
55 Since the existence of a subbase for a purely inseparable extension
is equivalent to the modularity of
L/K
L/K
in the
1.58 proves the existence of
bounded exponent case, Proposition
a maximal intermediate field with a modular base and a minimal intermediate field over which 1.59.
is modular:
L/K
K
Example.
= p(x,y,z),
, zP
-2
xP
-1
+ yP
P.
Then
{T .. } 1J
-1
x,y,z
)
where are independent
is a lower tower system
-1 -1 -2 -2 -1 T {zP , zP x P + yP }, T = {zP } and 2 2:= 12 -2 -1 -1 -2 -1 -1 h(zP x P + yP ), L/K Since i(ZP x P + yP )
where
L/K
¢.
TIl :=
-2
is a perfect field and
indeterminates over of
has a modular base.
Not every purely inseparable extension
Let L r K( zP
P
L
1.56.
is not modular by Proposition
The remainder of this section is devoted to the concept of modular closure as developed in [53J.
1.60. (a)
Lemma.
Let i
K(F P)
Land
F
L
K
be fields.
are linearly disjoint over
i
K(L L
n FP ), i:= 1,2, ... ,
n FP
F/L
and
i
and
K
is modular if and only if
F/K
are linearly disjoint for
is modular. i and (b) LP F/K
and
FP
i
i:= 1,2, •.. ,
and
nK
i := 1,2, ... , are linearly disjoint, i i is modular if and only if FP and K(L P ) are
linearly disjoint over is modular.
i
LP (K
n
i
FP ), i:= 1,2, ... ,
and
L/K
56
(c) K(L
n
i
FP ), i
disjoint, L
n FP i
LP (K
i
i
K(F P)
Land
i
i
1,2, ••• , i
K(L P )
and
n FP
1,2, ... , L P
=
=
=
), i
Proof.
are linearly disjoint over i
and
i
nK
and
FP
F/K
is modular if and only if
are linearly
are linearly disjoint over
1,2, ... , F/L
and
L/K
are modular.
This follows from the application of a well-known
property of linear disjointness (see Lemma in (24, p.162J) to the following diagram:
L
-----
K (L
i
K(LP )
K
___
FP )
.-----
----
---...
n
i
----------
. LP (K n F P )
K n FP
1
i ___
.>:
q.e.d. 1.61.
Lemma.
Let
F
L
K
be fields. is modular.
(a)
If
L/K
is separable, then
(b)
If
L/K
is separable algebraic, then
modular implies
(c)
If
F/L
F/L
L/K
F/K
is
is modular.
is separable, then
and only if
L/K
is modular.
F/K
is modular if
57
Proof. over
-i
Since Land KP are linearly disjoint i i i LP and Kover KP = LP n K.
(a)
K,
so are
(b)
Since
L/K
is separable algebraic,
c FP
and
FP
i
,
i
C
1,J+l -
T ..
f
L . . l(t) 1,J+
and by Lemma 2.10,
Lemma 2.10, using
e
t
where
t
E:
Since T .. },
1J
P
L.1- 1 , J+ . l(T.1 -1, J.)
we define
1-
1 , J.
P
f
to be
[t , p \ t P e: T . 1 . 1- , J
Hence
L i_ 1, j ... lub (Li_l, j+l (t' p)
I
Since
f
is a relative p-base, we have
that
L.1- 1 ,J+ . l(t )
f
1-
is 1
,
1
1 -
i-I.
and
T. l ' 1-
,
J
h
=
J N. . -
N.
1J
1
.\
l-,J
Li_l, j .
By the construction, there
satisfying (2) above with
i_ 1
That is,
. l(t' )}. 1- 1 ,J+
->
t ,p e T i-.r, j}
is a relative p-base.
. J
exists a mapping by
and
= I N1J
-
N
replaced
k 1
.\ •
l-,J
.. Then we define a 1J onto T . with the desired JJ ... ,T 1 j. Since the process is j,
This construction terminates with 1 -
1
mapping
g.
J
from
T .. JJ
properties with respect to
T
j
independent of the choice of defines a mapping of except that
[Tij}
N
onto
j,
the union of the
N'
g.'s J
with the desired properties,
was not an arbitrary lower tower system.
By Proposition l.42, there exists a composite mapping that has the desired properties when
{T . } 1J
is arbitrary.
q.e.d.
2.12.
Proposition.
Suppose
L/K
and
L'/K
are purely
inseparable extensions and (a) then
If
L'/K
L/K
has a lower tower generating system (set),
has a lower tower generating system (set) with the
same lower tower invariants. (b)
L/K
and
L'/K
have the same canonical invariants and
the same upper tower invariants. Proof. set of
J
(a) J-
1
(See Proposition 1.50.)
By Lemma 2.7, because
is a minimal generating
J
is one of
T.
J
L./L. l' JJ
conclusion follows here by use of Lemma 2.10 and
(b)
arguments similar to those used in Proposition 2.13.
Example.
When
L/K
and
L'/K
2.9.
q.e.d.
have the same canon-
ical generating invariants, their respective lattices
and
need not be isomorphic:
Let
-2
-1
L'
K(ZP
,zP
-2
yP
-1
+ xP
K )
p(x,y,z), L
where
P
K(ZP
x,y,z are independent indeterminates over P. -2 -1 tZP ,yP } is a modular basis of L/K, but L'/K
be isomorphic.
Now,
and
,YP-1 )
is a perfect field
and
have a modular basis.
-2
Then does not
Hence by Lemma 2.7, the lattices cannot tZP
-2
,yP
-1
}
and
[zp
-2
,zP
-2
yP
-1
+ xP
-1
1
are respective canonical generating systems with the same canonical invariants.
That
L'/K
does not have a modular base we know
by Example 1.59, but it also follows from (4) of Proposition 1.51, for -2 -1 -1 -2 (zp yP + x P )p C (L'P n K)(ZP )p). (L/K and L'/K also
84 have the same upper tower invariants.) 2.14.
Example.
When
L/K
and
L'/K
have the same lower
tower generating invariants, their respective lattices need not be isomorphic:
!'
K(ZP
-3
,zP
where
-3
P
xP
-1
+ yP
Let
-1-1
,wP )
K
P.
it follows that
L/K
elements while
Since
L'/K
yP
=
L'
and
is a perfect field and
terminates over
= p(x,y,z,w), K(ZP
x,y,z,w
-1
-3
-3
,zP
-3
xP
-1
+ yP
-1
=
-2 -1 -3 P -1 x + yP ,YP )
are independent inde-
is not needed to generate
has a minimal generating set with has one with
elements.
2
Lemma 2.7, the lattices cannot be isomorphic. [zp
,zP
L
-1
,wP }
and
and
!
{zp
-3
,zP
-3
xP
L'/K,
3
Hence, by
Now -1
+ yP
-2-1
,YP }
r P -3
_
are respective lower tower generating sets and T l.. Z , 33 -3 -1 -1 -2 -1 zP xP + yP }, T {zp }, T = {zp }, T22 = T12 = ¢, 23 13 -1 -3 -3 -1 -2 -2 TIl = and T = {zp ,zP + yP }, T = {zp }, 23 33
{wP } -1
{yP
}, T22 = Ti2 = ¢, Til have the same lower tower invariants. Ti3
=
xP
[zp
mapping
g
2.15.
-1
}.
Thus
L/K
and
In fact, there exists a
of the type described in Proposition 2.11. Example.
When
L/K
and
L '/K
L'/K
q.e.d.
have the same upper
and lower tower invariants, their respective lattices
s.
and
-4 need not be isomorphic: Let K = p(x,y,z,w), L = K(ZP , -4 -1 -1 -4 -4 -2 -3-1 p-3 zP x P , + yP ,xP ) ) and L' = K(ZP ,zP x P + y
!'
wP
where
P
is a perfect field and
terminates over
P.
x,y,z,w
are independent inde-
It is easy to verify that
L/K
and
L'/K
85 have the same lower tower invariants and the same upper tower invariants from the following lists.
fzP
-3
L
T == fzP 24 -2
-2
I.
T22 == fyP }, L/K and and
fyP
-1
respectively:
1.J
== fzP
fzP
-3
, zP
-4
-3
,zP
xP
have
},
-1
Mi4 == fzP
L'/K
have the following upper tower systems
fM . },
¢,
and
Ti4 ==
rwP }, M2 2 ==
L/K
-2
,zP
-2
M == fzP 14
-3
+ yP
-2
}, M
23
-4
,zP
-3
xP
M == fzP ,zP x + yP 24 -1 + yP }, M == ¢, M == 44 33
-4 P -2 -3 x + yP }, Mi3
-1
-4
== M
¢, Mil
Mi2 ==
22 ==
¢, M}4
== fzP
-2
-1
+ yP
},
}, and -1
==
-2
-3
,zP
-2
},
x
-1
M == ¢, M == fzP }. 44 33 -3 -3 -2 -3-2 Now K(L P) == K(ZP ,zP x + yP ) == K(ZP ,yP ) K(L'P) base
= K(ZP while
-3
,zP
-3 P -1 -2 x + yP ).
K(L,P)/K
does not.
mediate field lattices of
K(LP)/K
Hence
K(LP)/K
and
has a modular
Thus by Lemma 2.7 the interand
K(L,P)/K
are not
86 isomorphic.
Hence
B. More on type
and
R.
are not isomorphic by Lemma 2.10.
We now extend the investigation started in
I B.
2.16. of
L/K.
subset of
Definition. Set
S(L/K) == (L'
S(L/K).
only if for all
and
When
K
eeL)
for
2.17. lattice of S(L)
L/K
L'
£
Let
IL
be the intermediate field lattice £}.
I
Let
e(L/K)
is said to be of type
e(L/K),
L'/K
is of type
denote a
R(C) R.
is the fixed ground field, we often write S(L/K)
and
Definition. L/K. {L'
Let
e(L/K),
I L' e
Fc(L) == (L '
respectively.
be the intermediate field
£
£},
I L' e S(L),
L'
is distinguished in
[L: L'] < oo}.
we also define F(L) == (L ' IL'
S(L),
Bc(L) == (L' 1 L' c S(L), (L
I
Nc(L) == (L'
IL
I
S (L),
I L' e S(L),
[L
I
:
L/L' L'/K L/L'
K] < oo}, has an exponent}, has an exponent}, is modular },
1 L' e S (L),
L'/K
is modular 1,
U(L) == {L' 1 L' c S (L),
L'/K
has no exponent},
N(L)
(L
I
S(L)
In addition to the already defined sets
D(L) == (L' 1 L ' c S(L),
B(L)
if and
L/K},
87 UC(L)
:=
Uu(L)
:=
G(L)
:=
E(L)
I L' e S(L),
[L'
has no exponent},
L/L'
[L' I L' S(L), L/L' and -i [KP n L I i := O,l, ... }, i [K(LP ) I i := O,l, ••. J,
where the subscript
c
have no exponent} ,
L'/K
suggests cofinite, cobounded, comodular
and counbounded. Since R(B),
U(L) U B(L)
S(L)
and
L/K
we have by Corollary 1.18 that
if and only if R(U c)
:=
and
L/K
R(U u)
is always of type
L/K
has an exponent.
is of type
R(U)
Extensions of type
have an exponent if and only if relatively
perfect intermediate fields other than
K
are absent, as shown
by the Corollary to the following proposition. Proposition.
2.18.
Let
L/K
be purely inseparable.
Then the following conditions are equivalent:
if L'
(1)
L/K
is of type
R(U c) .
(2)
L/K
is of type
R(U u) .
(3)
For every intermediate field
L/L'
has no exponent then
Proof.
(1) implies (2) because
implies (3), let that of
L/L' L'/K.
is of type type
R.
L'
R. Hence
L"/K If
L/K,
has an exponent (that is
Uu(L)
E
Uc(L).
be an intermediate field of
has no exponent. If
of
L' e B(L)).
implies
Uc(L)
L'/K
L'
Let
L"
L'/K
L" e Du(L)
has an exponent, then
is of type
L/K
such
be an intermediate field
has no exponent, then L"/K
For (2)
R(S) .
Thus
L"/K L'/K
so is of has an
L"/K
88 exponent by Corollary (1) because by (3) type
Uc(L)
s
(Hence B(L)
Uu(L)
and
=
L/K
¢.)
(3) implies
is always of q.e.d.
R(B). 2.19.
K
1.18.
Corollary.
Let
L/K
be purely inseparable.
is the only relatively perfect intermediate field of
If L/K, then
(1), (2) and (3) of Proposition 2.18 are each equivalent to L/K
has an exponent. Proof.
nent. L)
L/K
Then there exists an intermediate field
of
L/K
not of type L'/K
Suppose condition (3) holds and
such that
L/L'
L'
(possibly
has an exponent and
L'/K
L'
:::>
Let
K(M).
have no exponent, otherwise e,
L'
=
L"
K(M).
L".
Hence
by condition (3) .
which contradicts the fact that perfect intermediate field of
K L/K.
Thus
L'/L"
Now
every
must
L"/K has an e+l e ), K(L'P ) = K(L'P
is the only relatively Hence
L/K
has an q.e.d.
exponent. If
is
M of
Then there exists a relative pbase
R.
such that
exponent, say
has no expo-
L/K
is of type
L' e S(L)
then
by Proposition
L/L'
1.15.
is of type Thus if
type
Rand
L/K*
is modular, then by Proposition 1.23
nent; hence
K*
R,
L/K*
is an intermediate field of
R
L/K L/K
L/K*
is of such that
has an expo-
has a modular basis by Proposition
The existence of a minimal
K*
is proved in Proposition 1.58.
such that
L/K*
for
1.56.
is modular
2.20. Suppose
Definition.
eeL)
L*
E:
If 1.20.
We say that
eeL)
such that
L/L*
is finite,
L*/K
is of type
[L },
then
eeL)
==
I
e S(L)
==
L/K
is of type
L'
Suppose and
Rm(e)
L'/K
semilattice in 2.22.
is of type
L/K
is of type
R(S) .
Let
is of type
Rm(e)
(b) Suppose that for
Rj(e)
is of type
if and only if
is of type base of
Since R
L/L'
R(e)
L/K
(a) Suppose
is cofinite in type
Then (1)
constitutes a L/K
is of type
L/K
L e e(L),
L/K,
(over
K).
Ll L
l
L/K
L' e S (LL L'/K
is
R.
Rm(e) .
is of type Let
L/K
is
is type
R(S) .
is of type L/K
eeL)
if and only if
is of type
by Proposition 1.15· and set
(a) Suppose
L e eeL).
L'/K
e S (L) .
R(e)}.
Set
constitutes a complete join-
finitely generated implies
L'
R( C) •
S(L).
L/K
Proof.
L(e)
and (2)
s(LL
L(e)
Proposition.
is cofinite.
is cofinite by Proposition
if and only if
complete meet-semilattice in if and only if
R(e) .
is cofinite.
F c (L)
Definition.
L(e)
K,
is cofinite if and only if and for all
2.21.
of type
containing
R(e)
In fact,
Rj(e)
L
is of type
such that
eeL)
fL'
as a fixed ground field.
L/K
L
S(L)
K
is defined for every field
eeL) S S(L). for all
Consider
R
Let whence
L/L'
M be a relative p-
n
L' (M - m). Since L'(M-m) meM is the intersection of fields of ==
Thus
Ll/K
is of type
R(e).
If
go L
L', then since LI/K I this process, replacing L
is of type by
LI.
R
we can continue
Hence there exists a
well-ordered set of intermediate fields of [L,L
L/L', namely
is
... }, each of type R(C) such that their intersection I, L'. Therefore L'/K is of type R(C) whence of type
R.
Hence
L/K
is of type
separable, then
L/K
R(S).
(If
is of type
Rj(C)
has an exponent by Corollary 1.18.) (b)
L'
L/K
is of type
Rj(C).
L'
lUb{K(a)
I a e L'}.
L'/K
is of type
R(C), L' L'/K
S(L),
L'/K
type
R(S).
of type
if and
L'/K
is of type
Suppose
is of type (If
Rm(C)
2.23.
L'
R(C).
L/K
K(a)/K
R(C).
L/K
Rj(B c)' Rj(F c)' Rj(E) Proof.
L'
which are composites
Hence we see that if
Rj(C)
if and only if
L L/K
is purely inseparable, then L/K
Suppose
and
Rj(N
C(L), is of L/K
has an exponent.)
L/K
has an exponent:
Each of the sets
an d each contains L.
is of type
Then, in particular,
The following types are identical, and if and only if
then
Conversely, suppose for all
if and only if
Corollary.
S(L),
for those
R(C).
L/K
The converse is obvious.
Since each
R(C)
of fields of type L/K
If
if and only if
is of type
is of type
is of type
then
S(L),
t
is purely in-
L/K
We first show that
only if for all
L/K
is q.e.d.
is purely inseparable. L/K
is of this type
Rm(B Rm(F Rm(E), c)' c)
c)'
Bc (L), Fc (L), E (L) are
cofini te q. e • d .
91 2.24.
Corollary.
The three types C
=
N, B, F
L'
R(C), Rm(C)
or
Proof.
Suppose and
is purely inseparable.
Rj(C)
are identical when
G.
For every subset
S(L),
L/K
L'/K
B'(L)
of
is always of type
B(L)
R(B').
and for every
Thus if
the intersection or composite of fields of type
L'
is
R(B' ),
then
is of type R(B'). Therefore the three types R(B' ), Rm(B' ) and Rj (B' ) are identical. If L/K is of type R( N),
L'/K
then
by Proposition 1.23.
N cB
Proposition.
2.25.
=
U c
U u
or
Proof. If L"/K
L"
Suppose
is purely inseparable.
L/K
R(C), Rm(C)
Then the three type C
q.e.d.
and
Rj(C)
are identical when
.
Suppose
L/K
is any element of is of type
was arbitrary,
R.
is of type Uc(L'),
Thus
of Proposition 2.22.
L"
Rj(U
L'
Let
S(L). Hence
Uc(L).
Since
R(U c)'
L'
as in the proof of (b)
c) and
Rj(U
are identical. c) By Proposition 2.18, the intersection of any collection of fields in
Uc(L)
nent over
K.
Suppose
L/K
type
S(L), Uu(L') R(U u)
= ¢.
Uc(L),
is of type
is of type
Proposition 2.18 whence L'
R(U c)
is again in
Hence L/K
Hence
then
is of type
L'/K
is of type
L/K
R(U c ).
L/K
R(U u)'
in fact of bounded expoRm(U c)' Then Uu(L)
is of type
Hence for all
L'
Rm(U u)'
=
¢ by
Also for all
S(L), L'/K
is of
so that in the proof of (b) of Proposition 2.22, q.e.d.
Reference Note for Chapter II The result 2.4 is due to Pickert [45J. invariants are treated in [15J, R
material is in [37J.
[17J.
The lattice
The additional type
Based on the fact that a proper
purely inseparable extension has non-trivial derivations, a Galois theory has been emerging;
some of the results of this
theory can be found in sources included in our reference list, in particular, [5J,
[13J, [22J, [24J, [55J, [57J.
III.
Some Applications
A. Extension coefficient fields.
We now consider for certain
commutative algebras the connection between our generating systems and the existence of coefficient fields containing the base field. Let
A
field g
be a commutative algebra with identity over the
A).
If
N
is a fixed maximal ideal of
is the natural algebra epimorphism of
identify
K
and
gK
in
A/N
A
onto
A
and if
A/N,
we
and sometimes denote
A
by
(A,K,N,g). We give a direct definition of the type of coefficiept fields with which we are concerned.
The well-known definition
and existence theorems in the case of complete local rings (see for example, [7J and [40J) do not in general imply the existence of our type. 3.1.
Definition.
coefficient field (or,
(A,K,N,g)
K-coefficient field)
and only if there exists a field and
is said to have an extension
F
in
A
F
for
N
such that
if
gF
=
gA
KeF. Even the tensor product
K(a) 0 F,
purely inseparable extension of exponent proper intermediate field of 1 0 F - coefficient field.
K(a)/K
where
K(a)
e > 1
and
is a simple F
is a
does not possess a
(Here and below,
0
means
0
K.)
94 3.2. H
Lemma.
K e L.
Suppose
H, K, L
There exists a subset
tive pbase of both
and
L/K
are fields such that
M of
L/H
L
which is a rela-
if and only if
=
K(LP )
H(LP ) . Proof. L/H(LP )
Suppose
M exists.
has exponent
1
K(LP ) = H(LP )
Then
K(L P ) £ L.
H(LP )
and
since
The converse q.e.d.
is immediate.
3.3. KeF
proposition.
L/F
Land
Let
K, F, L
has exponent
be fields such that
e.
Then the following
conditions are equivalent.
(1)
There exists an intermediate field
J/K
L/F
L
=
@K
as usual).
(2)
There exists a canonical generating system fBI" i i i P n K)(0:' ) where K. such that B!;> (L = a l l
of
L/F
F(B.l+ 1'" .,Be ), i
(3)
=
is modular and
such
L/K
that (@
J @ F,
of
J
is modular
.. ,Be}
1, ... ,e.
For all canonical generating systems
fBI, ... ,Be}
of
i
n K) (Kl L Proof. exponent one for
e
i
=
(1) implies (2):
1, ... , e.
Since
L
@ F,
J
J/K
and every canonical generating system of
L/F.
Since
generating system modular base of
J/K
is modular,
[Al, ... ,Ae}
J/K.
Clearly
J/K
such that [A
l,
... ,A
e}
also has
J/K
is
has a canonical
Al U ... U A is a e satisfies (2).
95 (2) implies (3): ating system of
L/F
Let
...
be a canonical gener-
satisfying the conditions in
(2).
Set
means F. Then where i == 1, ... ,e, Ki == F(B i+l,·· LP == Kl(Bl) so that (L P n K) (Kl) (Bl) == (L P n K)(Kl)' i i i Make the induction hypothesis that LP ee (LP n K) (Kl ). Then i+l i+l i+l i+l i+l i+l P ) == LP == (L n KP)(Kl ) E (LP n K)(Kl+ l i+l i+l i+l i (L P n K)(Kl+ l ) E LP • Hence by induction LP i i whence (LP n K) (Kl ) i == 1, ... LP
i
(LP
i
n K)(FP
i
i
i
), i
...
Thus LP
,
(L P
i
n
i F)(BI? 1, ...
Hence, by Proposition
i e
1.56, if
generating system of
), i == 1, ... , e.
tBi, ...
then i i
i
is any canonical
== 1, ... ,e.
is a relative pbase of BI?lU ... UBPe i i and LP /(LP n F) . By Lemma it follows that since i i i i IP U is a relative pbase of LP /(LP n it Bi+l U BeIP i i i i is also one for LP /(LP n K)(F P ). Thus LP == i i i i i i i (LP n K)(FP ... because LP /(LP n K)(F P ) i i i i i has an exponent. Hence BiP C (L P n K)(F P ..• ) Now
...
(LP
i
i
n K)(Kl ),
i
== i , ...
96 (3) implies (1): erating system of Assume
L
Let
fBl, ... ,B be any canonical gene} i i i Then LP = (L P n K)(Kl ), i = l, ... ,e.
L/F.
o K(A.l 1) 0 F(B., ... ,B) l e
K(Al) 0
tAl'" .,A.l 1,B., .•. ,B e } l
and
is a canonical generating system of i
i
of LP n K such that L/F. Take a subset Cl i i i i i KP (Ai, ... ,Al_l,cl)· Then LP i i i i i i i P ( AI'" P P P )( FP ) . Hence L = K .,Ai_l,Li ,BPi + l,·· .,BP e A U l is a relative pbase of L/F .
F(A l,·· .,A.l 1,C.,B. 1'" .,Be ). l l+
Select
... U A.
e
l
1 U A. U B. l
l+
1 U ... U B
A.l -c C.l
By the induction hypothesis it follows for all
so that
b
B. l
and for i
F* = F(A l, ... ,Ai_I' Bi + l, ... , Be) that [L: F*(B i b ) ] = p . Hence [B.} is a canonical generating system of L/F* with l
exponent fA.} l
i
i.
Since
A.l
is a minimal generating set of
is a canonical generating system of
L/F*
L/F*,
with exponent
by the invariance conditions of canonical generating systems. i
Thus, since
Al
E K,
L
K(A l) o ... 0
K(A i ) 0 F(B i + l ,
Therefore condition (1) holds by induction with (Also
L 0 Ki
has
K(Al,
efficient field, i = l, If
F/K
,A i)
o K.l
F/K
J 0 F
K(Al,
J/K,
thus for
L/K
,A e ) .
l
q.e.d.
is separable algebraic, then Proposition 3.3 not L/K
to be the tensor
of its maximal separable intermediate field
and the intermediate field of all purely inseparable
elements
).
10K. co
,e.)
only gives sufficient conditions for product
as a
=
J
,
but also criteria for to be modular by
(c)
J/K
to be modular, and
of Lemma 1.61.
97 We noted that (3) implies the condition:
(4) of
L/F,
For all canonical generating systems L @ K.
has
[Bl, ... ,Be}
1 @ K. - coefficient field, l
i
l, ... ,e.
=
Now (4) implies the following condition:
(5)
There exists a canonical generating system of
L/K
efficient field, Thus we have (1)
3.4.
C1L (2L
i
=
L
such that
e
K. l
has a
l, ... ,e.
(3)
(2)
Proposition.
(4)
If
F/K
(5). or
L/K
Suppose
F/K
or
L/K
is modular.
are linearly disjoint over
=
is modular, then
(3), (4) and (5) are equivalent.
Proof.
i
1 @ K.l - co-
Then
n
K
and
K,
1, ... , e.
We use this property to show that (5) implies (1).
Let
[Bl'" .,Be} be a canonical generating system of L/K which satisfies the conditions in (5) . Then L @F K has a coi efficient field containing 1 @F K. , i == 1, .. " e. We use the a symbol K e_ l
=
@ e F(B
to denote the tensor product with respect to e).
We now show
L/F
has a modular base.
Make the
induction hypothesis that cient field (i base
=
implies that
1, ... ,e)
for all extensions
has a
L @F Ki
L/F
L @ K. e
has a
l
has a modular
L/F
< e.
There
L @F K.l
onto
L
1 @ K. - coefficient field, e
coeffi-
with exponent
exists a canonical F-epimorphism of so
-
1 @F Ki
l
@
e K... l
i = l, ..... e - 1.
L/K has a modular base, say B'1 U . . . U B'e-l" by e_ l the induction hypothesis. We may assume that B U ... U B' e-l l' is canonically ordered. Since B , •.. , B:l - 1" Be satisfy the l'
Then
conditions in part B of Corollary 1.31, a canonical generating system of
L/F.
fBI" ... , B'e- l' B} e
is
L @F K l has e_ a 1 @F K coefficient field, there exist subsets B-lf -c e-l J j P that c 1 L @F K such f B-lf = and B-lf @F Ke-l e-l J J J (j = 1, ... , e - 1), where f is the canonical F-epimorphism of
L @F K e_ l
onto the residue field
maximal ideal. all where
b*
Bj
E: C
Now
BjP
j
it follows that
s eLand
e bI?
s J
Hence
c,p
j
j
elements of
Be'
b
s
By the division algorithm..
e
< p ,
pe,
rest) t
we have
where and
modulo its
j
@F 1)[1 @F
where
B:J+ 1 U ... U B'e- 1 U Be' b*P
L @F K e_ l
whence.for j j pJs pJ s '" p I t b*P = LJSC @F b ... b l t s
divides
E: F
l
j
b l, ... ,b t
nCs) + res), Since
S (LP
Because
i
=
1....... t.
it follows that where fb*P
c· e L. j
s
F(B e)
Also, with
n F(MjP
j
is a monomial in the
is a linear basis that includes
).
99
Thus each Since j
=
l, ... ,e - 1.
fBt ]
=
k
j
equals some
n j
C
]]
(L P
i
whence
j
n F)(M!P ), ]
{Bl', ... ,B ' 1,B} is a canonical gene e erating system satisfying (2) of Proposition 3.3 with K = F there.
Hence
We assume L 0
F
Hence
L/F
By, ... ,
of
L 0
such that
F
gBt1
L 0 i
e 1 0 K(FP ).
F
onto
=
LF
Since
g
L.
i
subsets
l, ... ,e,
=
and
is the canonical K-
Let
b
B!.
Then
By hypothesis there exists a subset
i
Ls s ms' where a s E: L i i i 6 s a Ps 0 mP = 6 s a P 0 (6 t s s a
b
where
k
and i
0
LP
st
= »,
{Ai, .•.
i
i
3.5.
m e F. s
)
=
n K and ms = 6 t kstXi
i k s t e LP .
i
Thus
(L
1
P
i
Also, Hence i
i
.
Suppose
are linearly disjoint over is modular.
i
= xP t
Hence for each i
n K)(F P ). L/F
i
,
t,
Therefore satisfying
3.3.
q.e.d. co
Lemma.
bP 0
is a canonical generating system of
(2) of Proposition
L/K
1
which is a linear basis of both i i i K(PP )/K. Now b P = 6 t k 0 { ,k e K. t t
and
kt
=
is a field, where
epimorphism of
FP
e
1 0 F coefficient field, there exist
(1 0 F)[B!, ...
of
U A'.
1
is canonically ordered.
A'1 U . . . U A'e
has a
A' U
has a modular base, say
n KeLP
i
) = K.
If
i=l r pi L n K(L P ), i < r
LP
i
and
1,2, ••. ,
r K(LP ) then
100
Proof. LP
that
i
Let
i r
K(LP)
and
intersection for subset of i
LP.
K
Then our hypothesis implies
are linearly disjoint over their i +
be a maximal P i) Pi which is linearly independent over L n K(L ==
X
Then
be fixed.
r ==
•
LP
K
X
is certainly linearly independent over each of
the smaller fields because
Let
r
i
== L,
+
i
.
are linearly disjoint and
r
K(LP ), X remains maximally linearly independent over
C
independent over i
LP
n K.
basis of
K/LP
basis of
r i K(LP )/LP
+ 1,... .
i
== .i ,
r
i
n
K.
Since
n
Also,
we show that LP
r K(LP ),
r
c LP
==
r
i
and there would exist a subset
mality of where
k
X.
Thus for any
r.
+
i
X U [x}
independent over
r
n
would not be in the linear span of
K
is clearly linearly
X
X
X X
is a linear is a linear if no t ,
...
over in
K
that is linearly
which contradicts the maxi-
k e
k
and
x r. e
==
j==O,l, . • • .
... ,
Hence
J
J
the coefficients in these linear combinations are in
n conclusion now holds because LP
i
.
X
K.
The desired
is linearly independent over q.e.d.
101
3.6.
Corollary.
Let
L/K
be purely inseparable.
n K(L P
(Xl
L/K
has a canonical generating system and
i
)
i=l
Suppose K.
Then the following conditions are equivalent. (1)
There exist subsets
A
l,A2,
every canonical generating system every positive integer and of
where
(3) i
(LP
c
-
L/K, L 0
i
L/K :=
K.
L/K
0
K(A e) 0
and K(Me)
is a canonical generating system
M == B e e+ l U Be+2 U
... .
i i n K) (Bl+ 1 ' Bl+2' . . . ) , i
has a
3.
::=
1 ® K.
3.
-
[BI' B2, ... )
of
[Bl' B2, ..• }
of
1,2, ...
coefficient field, where
1,2, . .. .
There exists a canonical generating system L ®
such that
K.
3.
has a
[B l,B
... } 2, 1 ® K.3. - coefficient field,
1,2, ... Proof.
Let
e
be any fixed positive integer.
any canonical generating system [B l, ... ,B
e}
F
Since
:=
of
For all canonical generating systems
K(M.3. ), i
(5)
i
K(A l) 0
such that for
}
For all canonical generating systems
(4 )
K.3.
l,B2,
L
.
L/K
L/K, Bl?3.
of
There exists a canonical generating system fBl' B ... } 2, i i i i such that Bl?3. E (L P n K)(Bl+ l,Bl+2" .. ), i 1,2, ...
(2)
of
L
[AI' ... ,Ae,Be+l,Be+2' ... } L/K,
of
e,
[B
...
K
e
•
[B
l,B2,
... }
of
is a canonical generating system of
i
i
n K)(Bl+ l,Bl+2' ... ),
Then for
L/K, L/Ke.
Set
we have the equivalence of (1), (2)
102
(3)
and
3.3.
by Proposition
(5).
implies
Now
(5)
(4)
Clearly (1) implies
implies a similar condition
which
(5')
in
e K(LP )
replaces K and the tensor product is with e e respect to K(L P ). Letting F == K(LP ) in Proposition 3.4, e we have that L/K(LP ) has a modular base. That is, (5' )
which
implies
e
L/K(LP )
Hence by Lemma
is modular for any positive integer
3.5,
proceed as in the proof of where we use the fact i
n
),
i
Setting
is modular.
L/K
(3)
L/K
e.
F::=K,
we
e
implies (1) of Proposition
3.3
modular implies that
= 1, ... ,e.
(1).
Hence we obtain (5) implies
q.e.d.
3.7.
Let
E/K
intermediate fields of
E/K,
Suppose e
Lemma.
beE
and
e
)n + kn_l(XP )n-l + .•. + k
rability of K
b
over
K.
and
i==O, ••• ,n
Proof.
and
S
separable algebraic.
in an indeterminate
o
where
Suppose Then
b
E:
n
over
is the degree of sepa-
S, F SF
x
are linearly disjoint
if and only if
p-e
k.
1
e F,
1.
Suppose
b
SF.
are linearly disjoint over K. e [K(bP ) : K] ::= n , Hence [F(b) S/K
S/K
F
is a root of the irreducible polynomial
0, ... ,n - 1,
over
be a field extension,
is separable algebraic,
Since Thus F] SF/F
bP
e E:
e S, K(bP )
and
F
e [F(bP ) : F] = e [F(b) : F(bP )]. n.
Since
is separable algebraic.
103 e b e F(bP )
Hence
[F(b): F] = n ,
and
'!herefore
b
is a
root of an irreducible polynomial of the form
+ ao Let F*
F*
over
F FKP
denote the composite
are linearly disjoint over
argument,
[F* (b) : F*] = n ,
where
-e
K.
in
a.1 c F, i = 0, ... ,n - 1. -e E(KP ). '!hen S
Similarly as in the above
'!h us it follows that
b
is a root -e
n p-e n-l of the irreducible polynomial q(x) = x + kn_lx + .•• + k oP -e over F* F. Hence k1? = a. e F, i = 0, ... ,n - 1. 1 1
-e
Conversely, suppose over
. Slnce
F.
separable over
x K
and
algebraic over
F.
K
q(x)
-e
Since
b
Hence
b
is separable
is also purely inseparable over q.e.d.
ElK
is finite,
purely inseparable over
F K
the intermediate field of all elements and
S
the maximal separable
intermediate field, then Proposition that
is irreducible and
SF. If
E
¢ S 0 F
sufficient for
ElF
implies that
exceptional extensions.
3.8.
q(x)
is irreducible and
=KP-e
, ...
is a root of
b
+ k n- IXn-l + -e
separable over
S, b
n
Then
F.
1
E
=S 0
Proposition.
intermediate field of
3.1 in [47, p.22l] shows always gives rise to
3.7 gives a necessary and
Lemma F. Let
L/K
L/K, and
denote a set of generators of
be algebraic,
F/K
Lis
S
maximal separable
a field extension. and for
b
t
G
let
Let
G
104
(x
pe)n + k _ ( x pe)n_l + ... + k o n 1
polynomial over e, n
K
which has
functions of
=
b, i
denote the monic irreducible
b
as a root
1, •.. ,n - 1).
unique composite and (2) for all if and only if (3) Proof. field.
=
LF LF
S 0 F
Suppose
For any
b
(1)
L/S
K
kl-e
beG,
and
LF
(2)
and
G, b
hold.
By
(1)
Then
l, ••. ,n - 1
is a
3.7.
Hence
Sand
F
in
(1)
LP
e
is purely inseparable and
S ® F
is a field, we q.e.d.
corollary. c Sand
FP
Suppose e
K,
a coefficient field of 3.10.
and let
f
f(S ® 1)
=
(2)
(in
k P.
-e
1
e
J
LF.
fel 0 S).
, i
=
LF)
FP
(2)
Proof. J ® F
If
00
then
K,
Let Let
L, K, S, F J
and
S ® F
G
is
Suppose Then (1)
L
n
F
L ® F
S
L ® F
Suppose (1), is a field.
for
Ker f
and
(2) and (3) hold. Thus
J
and
in
L/K
L ® F
LF
onto
and
has no nilpotent elements,
if and only if
1, ... , n - 1,
be defined
be an intermediate field
is a unique composite and
coefficient field of
imply
or
is a unique composite.
denote the canonical K-epimorphism of
a field composite
JF
3.8.
LF
L ® F.
Proposition.
as in Proposition
3.7.
Hence (2) holds by Lemma
K.
have that (1) holds.
3.9.
i
L 0 F.
S ® F
by (2) and Lemma
SF
k ,
is a
=
e F, i
is a coefficient field of
are linearly disjoint over
Since
t
i
Then (1)
conversely suppose (3) holds.
SF.
(k
F
(3) for all (4)
(5)
J 0 F L
beG,
is a
=
JS.
Then (1) and (2) (in
LF)
are
lOS
linearly disjoint over
K.
Hence
linearly disjoint over
K.
By Lemma
Thus
LF
JF
since
f(S 0 1)
(4) implies that over
K.
that
L/J
Hence (S) implies
J
L 0 F
S
J
(in
L)
are
(3) implies (S). Hence (4) holds.
f(l 0 S). and
F
are linearly disjoint
(3) by Lemma 3.7.
is separable algebraic.
L 0 J (J 0 F)
and
Since
Thus it follows
J 0 F
is a
has no nilpotent elements.
That
(1) holds.
q.e.d.
3.1l. and or
JP
(2)
has no nilpotent elements e e If ( 1) LP c S and JP =>
00
then
.:2
for If
J 0 F
is a coefficient field of
Ker f.
L/K
Proposition and
L ® F
Suppose
is a unique composite.
JF
L 0 F
Corollary.
is finite and
3.10 where
J/K
then
L
satisfies (S) of
is purely inseparable.
(3) of Proposition 3.10 hold.
Thus (2)
Hence (1) and (4) are equi-
valent in this case. If
J
is the intermediate field of
all elements of
L
3.12.
3.1 in
L/J
Proposition.
A
then
is not exceptional
p.221J). Let
be a complete local
algebra (not necessarily Noetherian). then
consisting of
which are purely inseparable over
(5) of Proposition 3.10 holds when (Proposition
L/K
If
A/N
is modular over
has an extension coefficient field if and only if
n
i
1,
. ..
.
106
Proof. i
Since
= 1,2, •..
p-base
B
of
A/N
A/N
i = 0
b
over
K
otherwise}.
Proposition 1.23.)
if
i
n K)
is modular over
such that
i = 1,2, .• . , c* = c n K nent of
g(AP
Suppose that
and
K,
there exists a
i K= «A/N)P n K)(C*), i is the expoC = [bP Ib e B, i
is purely inseparable over and
b
(See (*) in the proof of (2) implies (3) in
n
Since
i
K) = (A/N)P
n
K,
there
exists a set of representatives B' in A of B such that i AP [B' J contains c* (K and gK being identified) . Since i i i g(AP n K) = (A/N)P n K, we have K = (AP n K) (C*), i
1,2, . • . .
Thus
i
AP [B']
K, i
=
1,2, ... ,
whence with
co
respect to the
N-adic topology of
is a coefficient field of
A
A,
containing
n
i=l
(closure
K,
i
AP [B'J)
[58, p.306].
converse is immediate. When
The
q.e.d.
A/N
has no purely inseparable elements over K, i i then the condition g(AP n K) = (A/N)P n K always holds i pi since (A/N)P n K = K in this case. Also, if A/N is separable over
K,
then
A/N
is modular over
K.
The existence of an extension coefficient field is a particular case of the extension of a coefficient field of a subring to the whole ring [40J. this point of view.
We now give a few results of
107
3.13.
Proposition.
identity,
N
Let
A
be a commutative ring with
a maximal ideal of
homomorphism of
A
onto
A/N.
A Let
and R
g
the natural
be a complete local
ring (not necessarily Noetherian) of prime characteristic such that
R
A,
the identities of
A
and
M= R n N
is the unique maximal ideal of
R
R.
exists a coefficient field of A
A/N
RIM.
i
n RIM,
C
C
C'
(A/N)P
i
n RIM,
= 1,2, . . . .
i
i
RIM.
A
of
such that
B
is the exponent of
RIM,
is a p-base of
Suppose
g(AP
R
R
is a modular base of exponent over
K
over
A/N
that
gb'
is a coefficient field of Corollary.
b
= gb '
RIM and b' e B'
has over A.
Let
RIM,
B
- (RIM)
has the same
we see that
The converse is immediate.
K[B'] q.e.d.
A, N, R, M and g be defined as e e and gR = (A/N)P for some AP
then every coefficient field of
R
=
C'
Since
positive integer
A.
=
R)
has a coefficient field
If
extendable to one of
n
of
RIM} .
over
in Proposition 3.13. e,
B
i
where
C'
by the existence lemma as stated in [39J.
3.14.
Then there
Then there exists a set of repre-
1,2, ...
in
is
which is extendable to one of
is a p-base of
=
i
sentatives B' i {b'P b ' e B, Since
A/N
By Proposition 1.22, there exists a p-base
such that
(A/N)P
K
n R) =
if and only if Proof.
R
coincide and
Suppose
purely inseparable and has a modular base over
p
R
is
108
gK == (A/N)P for
(A/N)P
e
K
i
n
i g(AP n K) ::::: e K, i ::::: 1,2, ... , K(::::: gK::::: (A/N)P ). Clearly
K has a modular base.
over
Proposition
3.15.
Thus the conclusion follows from q.e.d.
Corollary.
Let
coincide.
Let
N
M M
be commutative rings
R
and the identities of
A
and
be maximal ideals of A and R, e e+l Suppose AP AP c R R n N.
for some non-negative integer R
and
A
A::J R
and
respectively, such that
(a)
A/N
3.13.
with identity such that R
e i-e «A/N)P )P
it follows that
e, e + 1, ... ,
i
Then
be a coefficient field of R. i e i Since (A/N)P n (A/N)P == (A/N)P
Let
Proof.
e,
P
:::::
characteristic of
R.
If
is a complete local ring (not necessarily
Noetherian), or e+l (b) AP
is a complete local ring (not necessarily
Noether ian) , then there exists a coefficient field of able to one of Proof.
e+i
(a)
Since
R
by Proposition
Corollary
R/M.
n (R/M), i
ficient field of E,
3.14,
which is extend-
A.
a modular base over (A/N)P
R
E
A/NP
e+l
S R/M
Now clearly
== 1,2, ...
e e (A/N)P , (A/N)P e+i g(AP n R) ==
Thus there exists a coef-
which is extendable to one of
3.13.
has
By identifying
E
and
say R
in
can be extended to a coefficient field of
A.
109
Since
(b) p-base and
e E R/M E (A/N)P ,
e+l
e such that (A/N)P e (A/N)P . Let G' E R
of
G UD
(R/M)(D)
(A/N)P
=
sets of representatives of
G
and
D,
there exists a
(A/N)P and
e+l
= R/M
(G)
D' CAP
e
be
respectively.
Now
e+l is a p-base of (A/N)P and e+l AP is complete, there exists a coefficient field e+l E[G' ] and AP such that E G'p U D'P. Hence K
Since E
of
E'
= K[D']
are coefficient fields of
3.14,
By Corollary
E'
Rand
e
AP,
respectively.
can be extended to a coefficient of
A.
q.e.d.
3.16.
Proposition.
unique maximal ideal
Let
A
be a quasi-local ring with
N and let
R
be a complete local ring
(not necessarily Noetherian) of prime characteristic that
R, M = R
A
coincide.
A/N R
p
N
If
e
n N and the identities of A and
=
for some positive integer
(0)
is separable over
RIM,
is extendable to one
A.
Proof. A/N
Let
K
where
B
A/N.
RIM,
RIM.
sets of representatives of e
if = (0), whence
L
is a field. e = AP [B',D'J
every p-base of
Hence let
is a p-base of
such R
e
and
then every coefficient field of
be any coefficient field of
is separable over
able to one of
P
BUD Let
Band
RIM
Since is extend-
be a p-base of
B' C K D,
R.
and
A/N
D' C A
respectively.
be
Since
Clearly
is a coefficient field of
A
as in
110
the proof of Corollary 3.14.
=
Now
L
=
e AP [B',D'J
e KP [B']
q.e.d.
K.
In Proposition 3.8 conditions, which include the case is purely inseparable, are given for
F/K
1 0 F - coefficient field.
existence of a
L 0 F
to have a
We now determine criteria for the
L 0 1 - coefficient field when
is purely
F/K
inseparable.
3.17.
Lemma.
Suppose either
Then there exists sets B U B L F
is a p-base of
Proof. set in
LF
L
=
(LF)P(F)
in
B* F
such that
BL U BF We may choose
=
F(LP).
e (L 0 F)P [B']
=
such that
is a p-base of
=
BL
B* L
and
Suppose
F/K
and
B* L
FP
e
.s
K,
(L 0
=
B F
and LF. of
LF. BF -c B* F
B F
or
BL
B* L
and
is a relative p-base of
has exponent
F)P
B* F q.e.d.
is a coefficient field of
Since
L(F P)
(LF)PCBt,B;)
L 0 F
is any set of a representatives of a p-base Proof.
F
which are p-independent
of
B L
B* so that either B or F L LF/F or LF/L, respectively. Lemma.
F
Thus
=
3.18.
in
(LF)P(Bt) = (LF)P(L)
Therefore there exist subsets
BF
B F
LF.
and a set
and such that
(LF)P(B;)
and
L
is algebraic.
F/K
It follows by Zorn's lemma that there exists a
in
B* L
in
B L
or
L/K
e C L 0
B 1.
e.
Then
where of
LF.
Thus
B'
III
e (LF)P.
is a field mapping naturally onto result by identifying
L ® F
with
A
We have the desired
and
(L 0 F)P
e
with
in the proof of Corollary 3.14.
3.19.
Proposition.
q.e.d.
Suppose
F/K
there exists an intermediate field L 0 F'
is a coefficient field of
B L
Land
of
LF'(= LF),
B F
of
F'
F'
has exponent of
L ® F,
such that
F/K
e.
If
such that
then for all subsets
B U B F L
e (1) L = K(LP )(B L),
we have that
R
is a p-base of (2) F' = K(B F)
e Conversely, if there (L ® F)P [B L ® 1,1 ® B F]. exist subsets B of L and B of F such that L F e (1' ) BL U BF is a p-base of LF, (2' ) L = K(LP ) (B L) and e then there exists an (L 0 F)P [B ® 1,1 0 BFJ, (3' ) K L
and
(3) K
intermediate field
F'
efficient field of
L 0 F,
Proof.
Suppose
of
L 0 F'
F/K
such that
namely
F'
=
K(B
L 0 F'
is a co-
F).
is a coefficient field.
By
Lemma 3.17, there exist subsets such that p-bases,
B U B L F
B of Land BF of F' L is a p-base of LF' = LF. For all such
e (L ® F)P [B
L
0 1,1 ® BFJ
is a coefficient field of
e by Lemma 3.18. Since (L 0 F)P [B L 0 1,1 0 BFJ L 0 F', e (L ® F)P [B L ® 1,1 ® BFJ = L ® F'. Thus K c e e (L 0 F)P [B L 0 1,1 ® BFJ whence K(L P )(B L) ® K(B F) e e = (L 0 F)P [B L o 1,1 o BFJ = L 0 F'. Hence L = K(LP ) (B L)
L 0 F
112 and
F'
K(B F).
=
hold.
Now
L 0 F
of
Conversely suppose (I'), (2') and (3') e
(L 0 F)P [B L 0 1,1 0 BFJ is a coefficient field containing K by (3'). Hence by (2')
e (L 0 F)P [B L 0 1,1 0 BFJ
=
e
K(L P ) (B L)
3.20.
q.e.d.
Corollary.
Suppose
F/K
there exists an intermediate field F'
0
L ®I«BF).
F' = K(B F).
Let
L
=
K(B F)
0
Bt U BF
Since
in
Land
is a p-base of
LF
and
K
(LF)P(Bt)
there exist BL U B F
and
Corollary. subsets
i
F/K
such that
BF
in
(LF)P(Bt)
= L whence
F
=
so that L(F P)
L
K, K
Suppose
F/K
has exponent
and
B F pe
B L
in LF
L and
(LF)
=
L.
= K(LP)(Bt).
FP
is a p-base of
tbP
of
Then
1.
L ® F.
Bt
3.21.
=
Choose
L 2 K(LP) (Bt) 2 (LF)P(Bt)
Then
F'
is a coefficient field of
Proof.
has exponent
(L 0 F)P(Bt ® 1).
in
F
(BV CF)
e.
q.e.d. If
such that
=
L
where
1b e
B i is the exponent of b over K}, then F, there exists an intermediate field F ' of F/K such that
CF
L ® F'
is a coefficient field of
Proof. Hence (L 0 F)
L
pe
L ® F.
We have
=
e K(L P )(B L)
(B
L 0 1,1 0 BF ] .
and
K
e
C
(L 0 F)P [B L ® 1,1 ® cFJ
(If it is assumed instead that
113
K
(LM)
C
pe
(BL,C F),
then
L
F
has a K-coefficient field.) q.e.d.
3.22.
Example.
field and L
=
K( c P
u, v
-f
=
K
p(u,v)
=
F
K(UP
-e
-e
,vP )
x)
If
f .::: e, then -f BL = fc P J and of
L
where
P
CF
over
K
B U B L F BF
=
=
and
e,
-e
is a perfect
c
F.
is a coefficient field of
L
F
When
If
(in an
LF
where is a p-base
BL U C F L
K(UP
f < e,
then
K(C)
by Proposition
L/K
Let
is a root of the
Thus by Corollary 3.21 L
P.
are positive integers [43J.
Clearly
}.
is a coefficient field of
B. Field composites.
f
is a p-base of
fu P
[u ] .
where
x 2 + ux + v
separable irreducible polynomial indeterminate
where
are independent indeterminates over
and
)
Let
-e
) F
3.8.
is purely inseparable and
F/K
is an arbitrary extension field, the analysis of the radical of
L
F
is useful for determining the structure of the
unique field composite
LF.
We now extend to fields which
have an exponent the basic methods and some of the results of the finite degree case [43J.
L/K
will be an extension
with exponent throughout this section. In the statement of the following proposition we use the simplifying fact that
F(LF)P
j
=
FLP
j
Also we omit the trivial
114 cases where one or both exponents are zero.
3.23.
Proposition. e
exponents
and
e'
,
chains of subsets of
and
LF/F
have positive
Then there exist two
L
:::J
:::J
M'
:::J
:::J
0
L/K
respectively.
M
0
Let
M.
1
J
:::J
:::J
:::J
:::J
M
:::J
e-l
M',
e -1
e =
¢
M'
=¢
M
:::J
e'
( 1)
such that i (a)
j
and
MJ? 1
i K(LP )/K
(i (b)
=
are minimal generating sets for
J
j
and
F(LP ) /F,
0, ... , e - 1; j
for each
j
respectively
0, ••• , e ' - 1),
and
there exists at least one
j'
such
that M.
J
Proof. by
A(S),
:::J
I
M
J
where
s
L,
J
+
1
(2)
j ' > j.
and
is an integer such that
s* ,
o -< s* -< e' ,
1 < s < e:
and two chains of
namely,
M e-s M'
:::J M. 1
:::J
e'-s*
such that:
M.,
Consider the proposition which we shall denote
There exists an integer subsets of
:::J
:::J
:::J :::J
-
M'
:::J
e'-l-
M'
e'
115
(1)
(2)
M.
The sets
and
l
j := e'-* 1) s , ... , e'-
have the properties (a)
and (b) of Proposition
3·23·
Either
s* := e'
s* < e'
F(LP
e'-(s*+l)
if
3.23.
and we are done, or p-independent in
is relatively p-independent in )/F
A(e)
i < e' - e* To prove
=
then
For, when e* < e'
F(LP
s
and
e'-(e*+l)
F(M o)'
M'
e'-s*+l
.1. M
r
e-s
is equivalent to the statement
is relatively p-independent in LF := F(K(Mo))
and
M', e -s *:= Me-s
First we note that of Proposition
or
and e'-(s*+l)
(ii)
M'e '-s* := M') 0 '
(hence
#e-s
( i)
Since
(i = e-s, ... ,e-l;
J
)/F.
=
e, then either e*:= e' e'-(e*+l) MP is relatively o
In the latter case,
F(L)/F.
M o
Since
we may therefore set
Mi
=
Mo
for
to complete the second chain. A(l)
we consider the set e-l is relatively p-independent in [I!I,SL, I P e-l is relatively p-independent in K(LP )/K, e'-l F(LP )/F}. e-l e'-l ) and F t F(LP K t K(L P ), there exist elements e'-l e-l aP t K and t F. If 1
116
aP 1
e'-l
{F
then
1
[a } E S1 ) ,
2
(a
+ a )P l 2
tal + a
2
}
aP 2
or if
e-l
f
K
then
But if neither of these negations hold,
e-l
f K
and
E Sfl),
(a
Thus
e'-l
+ a )P l 2
¢,
sfl)
f F,
then
which implies
and from the elementary
properties of relative p-independence it follows that is inductive. in
s(l) 1
Hence there exists a non-empty maximal set 1
say
'
1,
Suppose that
IP
and that
IP
e-l
p-base for
K(LP
p-base for
e'-l F(LP )/ F,
aP
e-l
)/ K
e K( Ii
1
e-l
1
e-l
is not a relative
e'-l
is not a relative
1
Then there exist elements
) t =
K( L
P
e ) (
Ii
e-l
)]
and
aP 2 If
aP
if
aP 2
e'-l
1
e'-l
f F( Ii e'
f F(LP )(Ii
e-l
e'-l
e '-1
) [= F(LP
)
1
1
these contradicts the maximality of 1 1 U tal + a
2}
either
can be chosen for
chosen for
M', 1 e -
1 < t for
< e'. M', l' e -
E sfl),
)( Ii
e'-l
)].
1 1 U tal} E sfl),
then
then
e'
U [a } 2
1
1,
E S
1 1
);
and
each of
But now we get
which is again a contradiction,
= ... = M', t e -
M
e-l
or if not then
for some maximal integer
If in the former case the set then we do so and set
s* = 1,
1
1
Hence can be t,
can be chosen
otherwise we let
117
s* s*
= =
O.
In the latter
e'
and construct a chain for
e- 1 -C
M' +c M
o the set
-c
•••
suppose
0
IP
Let
1
in
K(L P
in
F(LP
e-l
IP
)
e'-(t+l)
e-l
K(L P
relative p-base for
F(L
e'-(t+l)
e-l
)
I
K
pe'-(t+l)
Hence either
=
t)
case of and so
II:
Define
8(3) 1
just as in the case of
2
M
e-l s*
II
(and if
=
t +
M
1
t + 1 < e ,)
(if
and obtain
is attained M e-l as an extension of A(l)
for
t
then
if
1
M'
3
comes
0
is proved.
is true and consider
Me-(s+l) can be chosen as Me-s' true with (s + 1)* = s*. If
= ... =
is not a
then we obtain a con-
then
e-l M'0 + C M C C M Thus e-l - ... o' A(S)
is not a
can be chosen as 2 In the latter case we proceed as in the
until either
Nowassume
e-l
2
then we set
or if
then construct
M'o
)1
IP
e'-(t+l)
can be chosen as
at least.
then we consider
s
If IP 2
and
1
can also be chosen as s*
namely
is relatively p-independent
81 2 ) .
tradiction of the maximality of
otherwise
Then let
using
< e'
t
e'.
) IF}.
relative p-base for
above.
=
is relatively p-independent
be a maximal element of
2
L/K
But if
M •
t
A(s +
A(s +
1)
1).
If
is already
then suppose maximal.
Consider the set
118
=
e-(s+l) [11M E I E L, I P is relatively p-independe-s e-(s+l) e'-(s*+t+l) ent in K(LP )/ K, IP is relatively F(LP
p-independent in A maximal element of
is either an
Me-(s+l) A(l),
(c Me-(S+l)).
and an
M: P J
) / F}.
Proceeding as in the case
. Me-(s+l)
e'-(s*+t+l)
or an
we obtain an
Finally, we note n K(LP )/ K for
n
is relatively p-independent in n c L and is relatively p-independent n < j because J J n in F(LP )/F for n < j. Hence if j' is the largest nonthat
negative integer such that in
K(LP
j' ) / K,
3.24.
then
wP
j'
is relatively p-independent
J
with
M . '+1 eM: eM., J J J
Definition.
j' > j.
q.e.d.
The two chains (1) in Proposition
will be called compatible generating chains.
3.23
We associate with
such a pair of chains a pair of compatible canonical generating systems: and
B
J
where
=
B.
=
M. 1 - M.
M: 1 - M'.. JJ
3.25.
Remark.
If
L/K
is an m-fold purely inseparable finite
extension, then the original canonical description of this
[43J
extension ( a)
q. a.
(b)
aI?
(c)
e
-1
has the form q.
L
q.
eo:
e K(a l,·· .,a i_ l),
l > e 2 2:.
...
-> e m.
q.
=
p
e..i,
,
e. > 0,
119
When the finite degree case is thus expressed, Proposition
3.23
m'
tells us that there exists an integer where
ditions analogous to (a), the analysis of
LF
(b),
m' < m.
such
-'
satisfy con-
a l, ... , am'
(c).
[43J.
in
s
This fact is critical to
Note that the ordering of the
generators is the reverse of the ordering that arises naturally in the infinite degree case. To effect a reduction from the case where exponent to the case where
L/K
L/K
has an
is of finite degree, the
following proposition will be used.
3.26.
Proposition.
... LF/F,
Let
=
...
::l
M e_ l
and
be compatible generating chains for
respectively.
D
Let
there exists a finite subset L*
M o
K(B*),
be a finite subset of B*
of
M o
L/K Mo'
and Then
such that, with
we have
=
n
(a)
D
(b)
the finite degree extensions
L*/K
are canonically generated by
B*
C
B*,
L*F
F(B*
and
and
L*F/F B*
n M'o
respectively, where the ordering of the generators is that induced by
M o
and where the structure
of the canonical minimal polynomials of the generators is exactly the same in L*F/F Proof.
Let
as in
L/K
{Bl, ..• ,B
e}
and and
L*/K
and
LF/F. [Bi, ...
J
be the
compatible canonical generating systems determined by the
120
given chains.
Apply Proposition
maximal integers
such that
i. > j (j == 2, ... , e ' ) , J
-. Mi
-
3.23 as follows:
and
::J M'1 ::J M . J.
2- 1 -
-.
2 ::JM
D == Dl
and
J- 1
-
J
-
Now, set
M.J..-
There exist
::J
::JM.J.
e '-1 -
-
e'
-
1 -::JM', e - l'
and recall that e
M.J.- 1 ==
U .B n , i == 1, ... ,e,
n==J. e'
M: 1 == J-
U . n' n==J B'
We commence the construction of
j
1, •.. ,e'.
B*
by considering
There exist finite (possibly empty) sets
D , ••• , D.
2
J.
2
D n B 1. 1 such
Consider There exists a finite set
D' eM'
2 -
1
such that
Hence i
P 2
s; K(D i
2
-1 }.
We
121
such that
there exist finite sets
P K(D.
[(D 12 U D +1 U i2
1
(3) _
i
-1
3
- D12 U D + 1 i2
D12
such that
D' eM'
3 -
2
Since
we have
::) M.1 - 1
-
3
we have
Hence D 13
n B1
D 13
n B
n B1, ... , D
= D
12
i2
n B
= D
D i 2",·, 13
12
.,
,
B
2
i 2)P
Also,
(D
13
n B.1
n Bi
U Di -1)
3
3
3
= D
12
-
-1'
n
2-
1
and
1 Hence
i
2 S K(Dl +1)' .. ,,(D 13 2 2 2
2)P
B
=
number of steps yields
(Dr2
n
2)P
B
A similar process for
yields DIe' ... -- D'e = ¢, e' ::)M. cF because B', (DIe - DIe I)P e 1 e' De' ' +1 --
n B.1
... ,(D13
B1)P S i
n B.1 - 1 2
13
(D 12 U D.1 + 1 U 2
W1' th
Set
).
3
Now Set
B*
=
DIe'
122 Then
L*F
has canonical exponent of
n
F(B*) = F(B*
B*,
b
0
b E B* -
If in
LF/F
then
b
and by the construction
has canonical exponent
0
in
L*F/F.
The
conservation of the minimal polynomials follows from the fact that
(B*
n B.) J
pj
pj pj K( B. 1 7 ... 7 B ). J+ e
C
-
q.e.d.
For a given pair of compatible generating chains, the construction of
B*
from
we require that
D k
and
p
M.
1
i 7
i
=
B' o
1, ... ,e; j
M - M' o
=
k
be minimal.
K(Ml)/ K
and similarly for Set
will lead to a unique
D i
are possible because
D
0
B*
if
The latter choices
is minimally generated by
i
F(MiP)/ F. and call
07l7 ... ,e'.
B.
1
n
J
the i 7j-cell where
consider the cells as objects
ordered lexicographically by the lexicographical ordering of the pairs
(i,j).
Each cell as a set is considered to have a
fixed but arbitrary well-ordering imposed on it.
3.27.
Definition.
When the cells and their elements are
ordered as described above, we say that the minimal generating and
M'
o
are compatibly ordered.
E B. n and 1 a J then we sometimes write b
If
b
b(3 E Bk n B'P,
and
(i, j) < (k 7 -e L
i;
for B. n < Bk n B 1 a < b(3' J additional clarity we also use the notation b for b a, i, j a Suppose M0 , M'0 are compatibly ordered and let b E M . In 0 a Proposition 3.26 let us set D [b } and denote the uniquely a associated B* by [b }*. Then we spell out the latter set by a
.
123
where
n
depends on
b
and the elements are listed according a to the compatible ordering. Thus L*: K([b and a}*) L*F: K([b n The canonical degree of an element of B'o a}* is defined as 1 and is denoted by We shall often
abbreviate the canonical degree by using the notation when
b E M o
q' (b)
=1
and
when
Now, in
b
q'
(b)
f
B'o '
L*/K
when
b
Thus in particular
E M'.
o
as well as in
q(b)
L/K
we have the unique
expression b
q(b ) a a
f(x l,· .. ,X n ) E Kt x l, ... ,xnJ
where
Similarly, in
L*F/F
and
as well as in
deg x k < q(ba(k))/q(ba). LF/F, we have the unique
expression b
where
q'
(b )
a
a
(4 )
g(x l, ..• ,x n) f F[x l, ... ,xnJ
deg x k < q'(ba(k))/q'(ba). Let LF. C.l C
Of course,
=
M
order.
@
and
f
g
depend on
denote the canonical F-epimorphism of
We denote the radical
= B.l
and
@ 1, 1,
=
i
and
l, ..• ,e; C'
=
of l
M' @ 1. o
:
l
@ 1,
L i
F
q(c ) : q(b ) a a
F
N.
Set
onto
O,l, •.. ,e';
Order these sets by the induced
For clarity we also use the notation
Finally, set
by
L
and
c
..
a,l,J
for
r(c a ) : q(c a )/q'(c a ).
ba'
c . a
124 Clearly, (3) is satisfied in L 0 F,
after replacing
Let
F
L 0 FIN F
We use
C
b
by
as well as in
=
denote the following complete residue system for
c
as an algebra over i(c)
[I;FI1c
=
c
L* 0 F
F
(that is, over
1 0 F):
c ", 0.:::. iCc) < q(c)}.
IcE
F
to form our unique counter images with respect to c that is, Now suppose for some b that q(b ) = q' (b ), a a a terms omitted for brevity, r( ca ) = 1. Then, with q'(c) q'(c) f(Ca(l)a ... ) - g(ca(l)a ... ) r(c) > 1,
When
we define
a
w a
=
c
q' a
(c ) a
w
a
= O.
by (following Pickert)
q' (c ) g(ca(l)a , ... )
(5)
r(w )
w
a
where
g*
to the ba E
a
(6)
indicates that the coefficients of power.
r(c a) B
e n
r(w ) w a = 0, ) a a
I),
=
0
g
are raised
We note in particular, that if then
r(w ) w a = a
ifa
e
E 1 o F.
Since
in this case.
w defined by (5), a and order them as induced by our original compatible ordering. Denote by
W the set of elements
125
3.28.
Proposition.
The following set
power products is a linear basis of
N
S
of distinct
(over
F):
S := [IT c i ( c ) w j (w) IcE C ' , w E W, 0 < i ( c ) < q' ( c ) , 0
01.
Proof.
S'
Let
be a finite subset of
Each element
S.
of
S'
involves a finite subset of
of
W.
Since the defining equations for the relevant elements
of
W
involve only a finite subset of
all the finite subsets of Set
D:= cp(T)
Then
S'
case
[43,
F.
C'
is a subset of
and a finite subset
C' ,
the union
of
T
thus involved is itself finite.
in Proposition
3.26
L* 0 F,
p.IOOJ we find that
Finally we note that
over
C'
S'
to obtain
L*
and
L* 0 F.
so by the known finite degree is linearly independent over
Ker cp := FS,
so
S
generates
N
F.
q.e.d.
Since there exist only a finite number of cells, the following lemma is readily deduced from the finite degree case [42, p. 99J. w a
I
0,
Recall that
r(c
a)
> 1
if and only if
and that by (5): c
q' a
(c ) a
Hence, whenever a power of in a power product
c
a
occurs with an exponent IT ci(c),
where
c E C'
and
126
o
iCc) < q(cL
then this power of
c
(a) a power of
c
a sum involving only < q'(c
(b) wa ,
a),
and
c
(c) certain
can be replaced by
a
with exponent
a p
's
with
c
a < cpo
3.29. of elements of fixed
(i, j),
cells
>
n
with coefficients in
]
from cells
w
w
.
for a
]
can be expressed as a polynomial in
c's a
n
>
belonging to cells
C'
> B. l
n
If
]
F
(i,j)
c
w's
from
involving only is fixed and
then
"
a,l,J
r(w )
w
a
a
(8)
where
occurs with an exponent
< r(wa(s))' s == involve only in
w's
from
c's
coefficients of
••. , m,
from cells
which belong to cells
l
F
c
occurring elements of
n
c
and
n
]
< r(w)
F
and has coefficients
in each
w's
from the
wand with
involving (in addition to initially
C')
only
c's
which belong to cells
]
3.30. F
l
]
and c C. Every polynomial
can be expressed as a polynomial in
Fc '
coefficients from
1.
> B.
-
same set of cells but of degree
> B.
n
> B.
F a r e In a
Definition.
A polynomial expression in elements of
W of the final form described in part C. of the Lemma
3.29, is called reduced. is reduced.
In particular, every element of
F
c
127
It is readily shown that the reduced polynomial
(8) above has no term free of
w's,
F
of
a
which is indeed a necessary
and sufficient condition for a reduced polynomial to be an element of Let
N. i
be a positive integer. i N
notes, let of
denote the usual sum of products of
and let
N,
[6 F c
N i
Ni
Note that
In the remainder of these
N i
i
elements
be defined by
n wi(w) I w
E w,O < i(w) < r(w), 6 i(w) > L},
and N . a re F -mo d u 1 es.
Finally, let
l
W.
denote
l
the set of cosets defined by Wi == [n wi(w) + Ni+lJ w E W,O < i(w) < r(w),
with
W == W. o 3.31. N. l
and
Let
i
be a fixed positive integer. if and only if
N.l+ 1
as a LF-module. W
Proposition.
q'(b ) < q(b) a a
W. l
is a linear
In particular N/N2
is a linear basis of
minimal ideal basis of 3.32.
i},
The following is readily verified.
Remark.
if and only if
i(w)
2 N -- N2
(that is, a
N). If those
q'(b) 's
a
are all equal, then
N. l
for all
i.
for which
128 Proof.
By Proposition
3.26,
w
a
is an element of some
in which it maintains the relation (8).
L* 0 F
Since
L*/K
is of finite degree, we can apply the known finite degree result [43, p.lOl] to the radical
of
B* a
L*/K
N(B*) a denotes a minimal generating set for
3.26.
by Proposition i.
Clearly
and
N.
:0
1
N a
3.33.
I all
a Hence
1
as determined
We have that
U [N(B*)
U N(B*) ..
where
IT 0 F,
Corollary.
i N
N(B*)i:o N(B*). for all a a 1 i B* for all w }, N U N(B*)i a a a N.. q.e.d. :0
:0
1
Given the hypothesis of Proposition
3.32 we have (a)
lwl
(b)
Iwi
Proof. equal
q'
:0
I B o'1
if
E(b ) a
b
q.e.d.
is in the cell immediately less than that of The proof of
Sa tz 30
in
a
[43, p.l04J depends on the
finite degree case of the following theorem, which is an extension of Satz 29 in [43J of
N,
where
may write
ml
[N/N2: LF].
=
W = [WI' ... ,wn}
elements of
W.
and deals with the defect
Let
Suppose
Iw! =
n