Algebraic Theory of Numbers. (AM-1), Volume 1 9781400882809

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A LG E B R A IC THEORY OF NUMBERS

PRINCETON LANDMARKS IN MATHEMATICS AND PHYSICS Non-standard Analysis,

by Abraham Robinson

General Theory of Relativity,

by PAM , Dirac

Angular Momentum in Quantum Mechanics,

by A. R, Edmonds

Mathematical Foundations of Quantum Mechanics,

by John von Neumann

Introduction to Mathematical Logic,

by Alonzo Church Convex Analysis,

by R. Tyrrell Rockafellar Riemannian Geometry,

by Luther Pfahler Eisenhart The Classical Groups,

by Hermann Weyl

Topology from the Differentiable Viewpoint,

by John W, Milnor

Algebraic Theory of Numbers,

by Hermann Weyl

Continuous Geometry,

by John von Neumann

Linear Programming and Extensions,

by George B. Dantzig

Operator Techniques in Atomic Spectroscopy,

by Brian R, Judd

ALGEBRAIC THEORY OF NUMBERS

BY

H E R M A N N WEY L

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex Copyright © 1940 by Princeton University Press Copyright renewed © 1968 by Princeton University Press ISBN 0-691-05917-9 Sixth printing, 1971 First printing in the Princeton Landmarks in Mathematics and Physics series, 1998 http ://pup.princeton.edu

1 3 5 7 9

10 8 6 4 2

CONTENTS

Page Chapter 1. 2. 3. 4. 5. 6. 7.

I. ALGEBRAIC FIELDS ..................... Finite field. Norm, trace, discriminant . . Tower. Analysis of the field equation. . . Simple extension......................... Relative trace, norm and discriminant . . . Removal of the hypothesis of separability . The Galois c a s e ......................... Consecutive extensions replaced by a single one . . ................................ 8 . Strictly finite fields................... 9. Adjunction of Indeterminates.............

6 9 15 18 21 24 28 30

Chapter II. 1. 2. 3* 4. 5. 6. 7. 8. 9* 10. 11.

THEORY OF DIVISIBILITY (KRONECKER, DEDEKIND)......... ................. Integers ................................. Our disbelief in Ideals . . . . . ........ The axioms ............................... Consequences............................. Integrity inx(x,y,..) over k(x,y,..) . . . Kronecker's theory................. . The fundamental l e m m a ................... A batch of simple propositions........... Relative Norm of a Divisor............... The Dedekind c a s e ....................... Kronecker and Dedeklnd...................

1 1

Chapter III. 1.

2. 3. 4. 5. 6. 7. 8. 9. 10.

33 33 35 38 40 44 49 54 59 63 63 66

LOCAL PRIMADIC ANALYSIS (KDMMER, H E N S E L ) ........................... 71 Quadratic number field . . . ............. 71 Rummer*s theory: deconqposition.......... 75 Kummer’s theory: discriminant............ 79 Prime cyclotomic f i e l d s ................. 80 P r o g r a m ................................ 83 p-adic and £-adic n u m b e r s ............... 9^ and x ( £ ) ...........................100 Discriminant..................... . 107 Relative discriminant..................Ill Hilbert's theory of Galois fields. Art in symbol............................ . 116 v

Vi

ALGEBRAIC THEORY OF NUMBERS BY WEYL Page

Chapter III (Continued) 11. Cyclotomlc field and quadratic law of reciprocity............................ 124 50 12. General cyclotomic fields .................. 129 Chapter 1. 2. 5. 4. 5. 6. 7.

IV. ALGEBRAIC NUMBER F I E L D S ............ l4l Lattices (old-fashioned) .................. Field basis and basis of an ideal . . . . . . Norm and number of residues......... Euler!s function and Fermat!s theorem . . . . A new viewpoint................. Minkowski's geometric principle.......... 158 A fundamental inequality and itä consequences: existence of ramification ideals, classes of i d e a l s .............................. 163 8. The Dirichlet-Minkowski-Hasse-Chevalley con­ struction of units...................... 168 9. The structure of the group of u n i t s ...... 171 10. Finite Abelian groups and their characters . 11. Asymptotic equi-distribution of ideals over their cl a s s e s ............. 12. £ -function and related Dirichlet series . . . 13. Prime numbers in residue classes modulo m . 14. £ -function of quadratic fields, and their application............................ 193 15. Norm residues in quadratic f i e l d s ........ 201 16. General norm residue symbol and the theory of class f i e l d s .......................... 210

Amendment s

50 141 145 147 150 153

50

50 50 50 175 178 182 190

50 50 50 223

PREFACE

These are the authentic notes of a course on Theory of Numbers given In Princeton during the year 1958-1939; I say authentic because they were written down by the lecturer himself. The first two chapters of the course, dealing with the ele­ mentary theory of divisibility for ordinary inte­ gers and for polynomials of one and several vari­ ables, have here been omitted. Where I left off, Professor Chevalley continued with a course on Class Fields. In these notes, some of the materi­ al presented by him has been worked into the last chapter, on algebraic numbers, so as to pave the way to the modern theory of class fields and Abelian fields. In Chapter II I have axiomatized Kronecker:s approach to the problem of divisibil­ ity, which has recently been completely neglected in favor of ideals; the reasons for this procedure are given in the text. The ultimate verdict may be that the one outstanding way for any deeper penetration into the subject is the Kummer-Hensel p-adic theory. In view of the comparative scarci­ ty of books in English on theory of numbers, I hope that this outline of the fundamental arith­ metic concepts and facts concerning algebraic fields will be of some use.

Hermann Weyl The Institute for Advanced Study, Princeton, New Jersey

vii

A SHORT BIBLIOGRAPHY (BOOKS ONLY)

L. E. Dickson, History of the theory of numbers, Carnegie Institution, 1919-23, 3 vols. A. A. Albert, Modern Higher algebra, Chicago, 1937 v. d. Waerden, Moderne Algebra, Berlin, 1937 and 1931, 2 vols. H. Weber, Lehrbuch der Algebra, 2nd vol., Vieweg, 1Ö99 L. E. Dickson, Modern elementary theory of numbers, Chica­ go, 1939 Hardy and Wright, An Introduction to the theory of numbers, Oxford, 1938 Dirichlet-Dedekind, Vorlesungen Uber Zahlentheorie (4th and and last edition, Braunschweig, 18941) Algebraic numbers, Report of the Committee on Algebraic Numbers, Bulletin of the National Research Council, Nos. 28 and 62 (Washington, D.C., 1923 and 1928) E. Landau, Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, Leip­ zig, 1927 H. Minkowski, DiophantIsche Approximationen, Leipzig, 1907 E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, Leipzig, 1923 D. Hilbert, "Zahlbericht,M Gesammelte Abhandlungen, vol. I, Berlin, 1932, No. 7 (pp. 63-539) Hensel-Landsberg, Theorie der algebraischen Punktionen einer Variabein, Leipzig, 1902 K. Hensel, Theorie der algebraischen Zahlen, Leipzig, 1908 Krull, Idealtheorie, Ergehn, d. Math. IV 3, 1935 H. Hasse, Bericht Uber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Jahresber. Deutsch. Math. Ver. 35, 1926, 1-55; 36, 1927, 233-311 H* Hasse, Klassenkörpertheorie, Mimeographed Notes, Marburg, 1932-33 ix

Chapter I ALGEBRAIC FIELDS

I.

Finite

Field.

Norm,

Trace,

Discriminant

Let k be a field and k a subfield of x, so that h is a field "over k.M The elements of h are denoted by Greek letters and simply called elements, while Roman let­ ters' and the word "number” shall for the time being be re­ served for the elements of k. Elements may be added and an element multiplied by a number: these operations together with the axioms holding for them constitute x as a vector space. We assume that this vector space has a finite num­ ber n of dimensions; n is then said to be the degree of x over k, n = [x:k]. We repeat the well-known definition of dimensional­ ity for a vector space, and thus introduce the notion of a (vector) basis, m elements Ai,..., Am are linearly de­ pendent (with regard to k) If they satisfy a relation &3A 1 + .. . + amA.m = 0 with numbers ax,..., a m (In k) which do not all vanish, x is of degree n If any n + 1 elements of x are linearly de­ pendent while there exist n linearly independent elements coi, ••*, o)n. These form a basis, and every element | can be uniquely expressed in the form (1.1)

% = X 1 Ü)! + ... + x nun

where the^numbers xi,..., x n are called the components of £. Let o)i(i = 1,..., n) be any other basis. The 0)^ may be expressed in terms ofthe basis a)^, *

(1.2)

ü)

= ^T-ki^k*

and vice versa. Therefore the matrix L = ||1 ^ || must be non­ singular. Conversely if L = ||T-ik|| Is a non-singular matrix,

2

ALGEBRAIC THEORY OF NUMBERS BY WE2L

then (1.2) defines a new basis in terms of the original one ü)*. In expressing £ in terms of the new basis, I = JL%1 + ... +

we have

(1.3)

*i =

or in the notation of matrix calculus

_ *

x = Lx

with x standing for the column of the numbers Xi,..., xn . A linear mapping % — ► t\ In our vector space car­ ries the elementj 0)i of the given basis into elements and hence £, (1.1), into T1 - XiCDi + ... + xno>^. If

wi = £akiwk one has R = yi«Di + ... + y nü)n

with = E&ikXjc* Hence the linear mapping is described by a matrix A = ||aik|| in terms of the given basis. One easily verifies that the same mapping is described by the matrix (1.4)

L_1AL

In terms of the basis (w*) which arises from (o)j) by the transformation L, (1.2). We now take into account the operation of multiply­ ing any two elements of x. For a given element a the equa­ tion

n = a •%

ALGEBRAIC FIELDS

3

defines a linear mapping A: £ — ► r\ in x; A denotes at the same time the matrix expressing this mapping in terms of a given basis (ü)i): (1.5)

a • tüi =

A = ||alfc||.

The correspondence a — - A is a representation, called the regular representation, i.e., a — - A, ß — ► B entails ca —

cA (c any number),

a + ß — ► A + B,

aß —

AB.

Moreover the element 1 is represented by the (n-dimensional) unit matrix E. This is a good way of expressing the distributive and associative nature of multiplication. For instance, aß — - AB states that the mapping associated with aß is obtained by performing the two mappings associated with ß and a, one after the other (first ß, then a); or (aß)| = a(ß|). While we started by considering x as a vector space (first level) it now appears more particularly as an alge­ bra (second level). The peculiarities which characterize a field (third level) among algebras are (1) the axiom of division: for any a 4 0 there exists an element a “1 such that a • a ^1 = l (division algebra); (2) the axiom of commutativity of multiplication. They are of a considerably more refractory nature than the assumptions characterizing x as an algebra. For the time being we continue to move on the second level. Let A be a linear mapping and its matrix, ||ajjcJ, t an indeterminate. The characteristic polynomial (1 .6)

f(t) = tn - ait11"1 + a2tn~2 - ... + an

of the mapping A is introduced as the determinant of the matrix tE - A. According to (1.4), f(t) is an invariant of A, namely independent of the basis in terms of which the mapping is expressed as a matrix A. In particular the trace ai = S a n and the norm, an = det(aik)

ALGEBRAIC THEORY OF NUMBERS BY WEYL

k

are Invariants. B equals

A,

The trace of the product of two mappings

and hence is symmetric In A and B. We apply these remarks to the linear mapping A: % — - a|, associated by the regular representation with the element a. The trace and norm of A are called trace and norm of a and denotedby S(a), respectively. The

Nm a

equations (1.5) or £(a6ki - aki)(% = 0

at once show that det(a6kl - akl) = 0, or that a is a root of the characteristic equation f(t) = det(tE - A ) . [E = || || denotes the unit matrix.] We therefore call f(t) the field equation of cl, and we have proved: Theorem I I, A . E v e r y e l e m e n t o f x s a t t s f i e s a d e f i n i t e a l i e b r a i c e q u a t i o n o f d e i r e e n with c o e f f i ­ c i e n t s i n k, i t s f i e l d e q u a t i o n . The

trace S(£) depends linearly

S(a

+ß) =

S(a) + S(ß),

on

S(ca) = c • S(a) (c any number in k).

Moreover 3(1) = nl. The norm has the multiplicative property (1.7)

Nm(aß) = Nm a • Nm ß ,

ALGEBRAIC FIELDS

5

and for any number c In k Nm(c) = cn . One Is tempted to write the characteristic equa­ tion f(t) as Nm(t - a), and this Is indeed justified if one replaces k and x by the rings k[t], x[t] of polynomials of t with coefficients in k and x respectively. x[t] is of degree n with respect to k[t], and the basis coi,..., ü)n of x/k is also a basis for x[t] relative to k[t]. Later on (§9) ve shall study the adjunction of Indeterminates more systematically. An equation like Nm(t - a) = f(t) stays true if one substitutes for t a number ink; substitution for t of an element in x, however, is strictly forbidden! One effective way to make use of x being a division algebra (third level) is by the fact: a * 0 implies Nm a 4 0. Indeed, with the inverse a "1 of a * 0, aa“1 = 1 , one infers from (1.7)2 Nm(a) • Nm(a-1) = 1. The trace S(£t]) is a symmetric bilinear form of the two variable elements £, “H, which we call their scalar product. Expressing it in terms of a given basisw*, £ = |Xiü)i,

q =

one gets (1.8)

S(£q) = Eaj.fcXijqr

with the coefficients

s ik =

•

Our field is said to be non-degenerate, if this bilinear form is non-degenerate, I.e., if a = 0 is the only element for which the equation S(aT|) =* 0 holds identically in tj.

6

ALGEBRAIC THEORY OP NUMBERS BY WEYL

This means that the discriminant of the basis Wi, namely D(u>i,..., o)n) = det(slk) is different from zero. Under the transformation (1.2) of the basis, the determinant of the invariant form (1 .8) as­ sumes the factor |lit]2,

“ n) = |l-ik |2 *

con).

In a non-degenerate field the discriminant of no basis van­ ishes while in a degenerate field the discriminant of every basis vanishes. If a 4= 0 one has S(a|) = nl

for

| = a-1.

Hence our field x is certainly non-degenerate unless nl = 0, 1.e., unless k is of a prime characteristic dividing the degree [x:k]. 2.

Tower.

Analysis

of

the

FieldEquation

"Finite field (over k)" is used as a shorthand term to describe fields of finite degree over k. Theorem 1 2 , A . I f x i s a f i n i t e f i e l d ( o v e r k) o f d e g r e e n and K a f i n i t e f i e l d o v e r x o f d e g r e e r, t h e n l i s a f i n i t e f i e l d ( o v e r k) o f d e g r e e N = n • r: (2 .1)

[K:k] = [K:x] • [x:k].

We speak of r as the relative degree of K/x while n and N are the flab solute11degrees of x and K respectively, k isconsidered as the ground level on which our structures rise, and Mabsolute” therefore means the same as ”relative to k.” At the present stage we apply the word "number” in­ discriminately to the elements of k and of the fields over k. Using x as ground field over which to erect a super­ structure like K is an effective means of availing oneself of the commutative nature of the division algebra x. The proof of our theorem is very simple indeed. Let &b(s * 1,..., r) be a relative basis of K/x so that each number S of K is uniquely expressible as a ~

8=1

(Ss

H) •

ALGEBRAIC FIELDS

7

and let i(i = 1,..., n) be a basis of x in terms of which the coefficients |s in their turn are expressible: n ?S = J ^ x i b W i . One then realizes that the numbers (2 .2)

a • am-1 = -bm - ... - biam-1 as the sought-for polynomial t, 0, 0, -1, t, 0, 0, -1, t, 0,

0, 0,

bm bm-1

bm- 2 t + bi

In replacing the first row by the linear combination of the first, second, ..., m^*1 rows with the coefficients 1, t,..., t111“1, one arrives at the result wanted. x must be a field over k(a) of a certain relative degree r, and by applying our former results to the tower k c k(a) ßs)

*

Erratum for second printing. This sentence should be in Italic type instead of Roman.

50

ALGEBRAIC THEORY OP NUMBERS BY WEYL

equal If every integer divisible by (Jl Is divisible by & , and vice versa. (Jl is said to be divisible by , (Jl \ , if every number divisible by (Jl is divisible 'by Jx . Hence (Jl = £ if i J r and £ : (Jl. Our definition may be stated thus: a is divisible by Ä If a is divisible by the linear form axxi + ...+ arxr. In this sense the linear form may stand for the divisor (a i f . . . y a r)• Using the criterion (5.2) we can put the criterion for divisibility of an Integer a by the divisor (Jl = (o-i,..., oir) into this neat form:

Ol

(6.4) Ct Nm(ax + a xxi + ... + arxr) : Ct Nm(a!Xi + ... + arxr) •

(Jl

Whether a Is divisible by can therefore be decided, pro­ vided we are able to decide in the ground field k whether the GCD of certain numbers (ai, a2,...) is divisible by the GCD of certain other numbers (bi, b2, ...). Theorem II 6, A .

i s d i v i s i b l e by (JL = Proof.

Each o f the numbers a a rJ.

a l f ... , ar

(

We show that

(6 .5)

cciyi + ... + aryr aixi + ... -I- arxr

for yx = l, y2 = ... = y r = 0 is an integer In x(xi ... xr). With the homogeneous form f(x.i ... xr) = Nm(aixi + in k of

... + arxr)

degree n one obtains as the field equation of (6 .5): f(tXi - yi, ..

txr - 7r ) '

f(xi,..., x r ) It is evident that the GCD of the numerator is divisible by that of the denominator. Theorem II 6, B.

i f- and onl y i f

(Jl i s d i v i s i b l e by & , ( 6 . 3 ) ,

a l * l + • • • + 0Lr Xr ..., ßs ) is a common divisor of ax,..., a r, then (6 .7) is integral in x(yi... ys) and (6 .6) in x(xi ... xr, yi ... ys), or Ol : ;£ . Theorem II 6 , D.

namely (6. 9)

Two d i v i s o r s (6. 3) have a GCD,

(OL. & ) = f a ! , . . . , d r , pl f Proof.

Pe ; .

The argument In Theorem II 6, A, proves

Q-ixl + ... + CLrXr + ßiYl + .*. + ßsYs ttiXi -I- ... + arXr + ßiYi + .•• + ßeYs

ALGEBRAIC t h e o r y o p n u m b e r s b y w e y l

52

to be integral with yl = •.. = y B = 0, or Ä ble by (6,9). Vice versa, if fciXi + ... + arx r YiZi + ... + Ytzt *

to be divisi­

ßiyi + + ßsYe YiZi + ... + Ytzt

are integral in x(x,z) and x(y,z) respectively, then their sum is integral in x(x,y,z); or with Ql and divisible by £ , (6 .9) is divisible by £ . Multiplication of two linear forms cxiXi + ... + arXr

and

ßiyi + ... + ßsYs

yields a bilinear form (i = 1,..., r; k = 1,..., s) Hence in order to define the product of two divisors (6.3) it seems necessary to go beyond the bounds of linear forms. Fortunately this is not so because (6.10)

(Saixi) (Eß^yt) ~ Ea-ißkuik

with rs indeterminates u ^ . that

Indeed, It follows from (6.5)

Saißkx^k 2a ißku ik Is Integral, and on the other hand, by multiplying ___________ axxi + ... + arxr 9

ßk_______ ßiyi + ... + ßBys

one realizes that a-ißfc and hence Eaißkuik Is divisible by the left member of (6.10). We therefore define: An inte­ ger Y is divisible by O llr if _______ Y_______ Zaixi • 2ßkyk is integral in x(x,y), and then observe that 0 l £ Is the divisor £

- (..., cxißk* ...)

[i = 1,..., r ; k * 1,..., s]

THEORY OF DIVISIBILITY (KRONECKER, DEDEKIND)_______ In the sense that every integer divisible by ÖL& Is divis­ ible by