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B.L. van der Waerden

Algebra

Volume II

Based in part on lectures by E. Artin and E. Noether

Translated by John R. Schulenberger

Springer

PREFACE TO THE FIFfH EDITION

P. Roquette has been kind enough to provide me with a nice proof of

the residue theorem for algebraic differentials udz. The- chapter "Algebraic

Functions" has thereby been brought to a satisfactory conclusion.

In the chapter "Topological Algebra," following Bourbaki, the completion of

groups, rings, and fields has been carried out by means of filters without using

the second countability axiom.

The chapter "Linear Algebra," which is important for many applications, now

appears at the beginning of the volume, and topological algebra is treated in the

last chapter. The book now consists of three independent groups of three

chapters each:

Chapters 12-14: Linear Algebra, Algebra, Representation Theory

Chapters 15-17: Ideal Theory

Chapters 18-20: Fields with Valuations, Algebraic Functions, Topological

Algebra

This subdivision of the material is now expressed more clearly in the schematic

guide on page xv.

Zurich, March 1967 B. L. VAN DER WAERDEN

FROM THE PREFACE TO THE FOURTH EDITION

Two new chapters have been added at the beginning of the second volume:

a chapter on algebraic functions of one variable, which goes as far as the

Riemann-Roch theorem for arbitrary fields of constants, and a chapter on

topological algebra, which is mainlyconcemed with the completion of topological

groups, rings, and skew fields. I should like to thank Dr. H. R. Fischer, who

read these two chapters in manuscript form, for many useful remarks.

The chapter "General Ideal Theory" has been extended to include the impor-

tant theorems ot Krull on symbolic powers of prime ideals and chains of prime

ideals. The relation of the ideal theory of integrally closed rings with valuation

theory has been brought out more clearly. A section on antisymmetric bilinear

forms has been added to the chapter "Linear Algebra."

In the chapter "Algebras," more examples are given, the theory of the radical

has been developed, following Jacobson, without a finiteness condition, and

the fundamental ideas of Emmy Noether on direct sums and intersections of

modules have been more strongly emphasized. It was possible to considerably

simplify the proofs of the principal theorems by combining the methods of

Jacobson with those of Emmy Noether.

By omitting some material I have tried to keep the size of the book within

reasonable bounds. Thus, the chapter "Elimination Theory" has been omitted.

The theorem on the existence of the resultant system for homogeneous equations,

which was formerly proved by means of elimination theory, now appears in

Section 121 as a corollary of Hilbert's Nullstellensatz.

Zurich, June 1959 B. L. VAN DER W AERDEN

CONTENTS

14.1 Statement of the Problem 14.2 Representation of Algebras 14.3 Representations of the Center

Valuation of Algebraic Extension Fields: Complete Case

 - Chapter - LINEAR ALGEBRA 

• 12.1 Modules over a Ring
• 12.2 Modules over Euclidean Rings. Elementary Divisors
• 12.3 The Fundamental Theorem of Abelian Groups
• 12.4 Representations and Representation Modules
• 12.5 Normal Forms of a Matrix in a Commutative Field
• 12.6 Elementary Divisors and Characteristic Functions
• 12.7 Quadratic and Hermitian Forms
• 12.8 Antisymmetric Bilinear Forms - Chapter
  • ALGEBRAS
• 13.1 Direct Sums and Intersections
• 13.2 Examples of Algebras
• 13.3 Products and Crossed Products
• 13.4 Algebras as Groups with Operators. Modules and Representations
• 13.5 The Large and Small Radicals
• 13.6 The Star Product
• 13.7 Rings with Minimal Condition
• 13.8 Two.-Sided Decompositions and Center Decomposition
• 13.9 Simple and Primitive Rings
• 13.10 The Endomorphism Ring of a Direct Sum
• 13.11 Structure Theorems for Semisimple and Simple Rings
• 13.12 The Behavior of Algebras under Extension of-the Base Field - Chapter - REPRESENTATION THEORY OF GROUPS AND ALGEBRAS - 14.4 Traces and Characters X CONTENTS - 14.5 Representations of Finite Groups - 14.6 Group Characters - 14.7 The Representations of the Symmetric Groups - 14.8 Semigroups of Linear Transformations - 14.9 Double Modules and Products of Algebras
• 14.10 The Splitting Fields of a Simple Algebra
• 14.11 The Brauer Group. Factor Systems - Chapter - GENERAL IDEAL THEORY OF COMMUTATIVE RINGS - 15.1 Noetherian Rings - 15.2 Products and Quotients of Ideals - 15.3 Prime Ideals and Primary Ideals - 15.4 The General Decomposition Theorem - 15.5 The First Uniqueness Theorem - 15.6 Isolated Components and Symbolic Powers - 15.7 Theory of Relatively Prime Ideals - 15.8 Single-Primed Ideals - 15.9 Quotient Rings
  • 15.10 The Intersection of all Powers of an Ideal - Rings 15.11 The Length of a Primary Ideal. Chains of Primary Ideals in Noetherian - Chapter - THEORY OF POLYNOMIAL IDEALS - 16.1 Algebraic Manifolds - 16.2 The Universal Field - 16.3 The Zeros of a Prime Ideal - 16.4 The Dimension - 16.6 Primary Ideals - 16.7 Noether's Theorem - Chapter (7ontents xi
    • INTEGRAL ALGEBRAIC ELEMENTS
• 17.1 Finite 9t-Modules
• 17.2 Integral Elements over a Ring
• 17.3 The Integral Elements of a Field
  • 17.4 Axiomatic Foundation of Classical Ideal Theory
• 17.5 Converse and Extension of Results
• 17.6 Fractional Ideals
• 17.7 Ideal Theory of Arbitrary Integrally Closed Integral Domains - Chapter - FIELDS WITH VALUATIONS
  • 18.  - Valuations 

        • Complete Extensions
          • Valuations of the Field of Rational Numbers
          • Valuations of Algebraic Number Fields Valuation of Algebraic Extension Fields: General Case
          • Valuations of a Field d{x) of Rational Functions
          • The Approximation Theorem
    • Chapter
    • ALGEBRAIC FUNCTIONS OF ONE VARIABLE
  • 19.1 Series Expansions in the Uniformizing Variable
• 19.2 Divisors and Multiples
• 19.3 The Genus g
• 19.4 Vectors and Covectors
• 19.5 Differentials. The Theorem on the Speciality Index
• 19.6 The Riemann-Roch Theorem
• 19.7 Separable Generation of Function Fields
• 19.8 Differentials and Integral~ in the Classical Case
• 19.9 Proof of the Residue Theorem - Chapter xii CONTENTS - TOPOLOGICAL ALGEBRA - 20. - 20. - 20. - 20. - 20. - 20. - 20. - 20. - 20.

• 20.  - The Concept of a Topological Space - Neighborhood Bases - Continuity. Limits - Separation and Countability Axioms - Topological Groups - Neighborhoods of the Identity - Subgroups and Factor Groups - T-Rings and Skew T-Fields - Filters Group Completion by Means of Fundamental Sequences - Group Completion by Means of Cauchy Filters - Topological Vector Spaces - Ring Completion - Completion of Skew Fields 

  • Index

2 LINEAR ALGEBRA

factors form a submodule IDtl for which B is the identity operator. These two

submodules have only the zero element in common, since for any other element

annihilation and reproduction are mutually exclusive. The representation (12.1)

shows, moreover, that IDl is the direct sum mo +IDll • After the uninteresting part

IDlo ofilR is split off, we obtain a module for which B is the identity operator. We

shall therefore assume in the following that the identity element of m is also the

identity operator for IDl.

If, in particular, 9t is a skew field, then IDl is a vector space over 9t in the sense

of Section 4.1, Volume I.

The module IDl is said to befinite over 9t if its elements can be expressed linearly

in terms of finitely many basis elements Ul' ••• , u,.:

UtAl + · · · + u,.A,.. (12.2)

In this case IDl is the sum of the submodules u 1 9t, ... ,ullm:

• IDl = (u (^) l 91, ..• , U,.m). (12.3)

Instead of (12.3) we sometimes write for brevity:

IDl = (U1' •.• , un).

Ifin the representation (12.2) the coefficients AI, ••• ; An are uniquely determined

by u, then 9R is called a module of linear forms over 91. In this case the sum (12.3)

is direct:

IDl = u 1 91+ · · · +~nm.

Every finite-dimensional vector space is a module of linear forms, since by

Section 4.1 we can always choose a linearly independent basis (U1' ••• , u,.). By

Section'4.2 the dimension n is independent of the choice of basis.

An operator homomorphism which maps a module of linear forms IDl =

(U1' •.. , um) into a module of linear forms m= (vI' •.. , VII) is called a linear transformation ofIDl into m. For such a transformation A, therefore, we have, as

in Section 4.5,

A(x+y) = Ax+Ay

A(XA) = (Ax)A.

The transformation A is completely determined if the image of each basis

element Uk'

AUk = L U'(Xii'

is given. The coefficients (Xile form a matrix of the transformation A..

If A is a one-to-one mapping ofIDl onto 91, then there exists an inverse mapping

A -1. We then have

and AA- 1 = 1,

where 1 denotes the identity. In this case the mapping A and its matrix (<X,i) are

called invertible.

Modules Over Euclidean Rings. Elementary Divisors 3

In the following we shall often denote the linear transformation A and its matrix (a:i1J by the same letter A. This is not altogether logical, but it is practical.

12.2 MODULES OVER EUCLIDEAN RINGS. ELEMENTARy DIVISORS

We now require that the ring 9t be commutative and Euclidean in the sense

of Section 3.7. This means that to every ring element a =t= 0 there corresponds an "absolute value" g(a) such that g(ab) ~ g(a) and also that a process of division

is possible. According to Section 3.7, every ideal in 9l is then a principal ideal.

Theorem: LetIDl be a module o/linear forms over 9l with basis (u 1 , ••• , u,.). Then

every suhmodule 9l ofIDl is again a module of linear forms with at most n basis

elements. Proof: For the null module IDl = (0) the theorem is trivial. Suppose then that it is true for modules 9R with n -1 basis elements.

If 91 consists of linear forms in U1' ••• , U" -1 only, then the theorem is true by

the induction hypothesis. If mcontains a linear form u 1 A 1 + .. · +U"A" with

A" =t= 0, then the set of such A" forms a right ideal in 9t which is thus a principal

ideal (p,n) with 11-" =l= o. Therefore 9l contains a form I = U 1 fLl + ... U"IL,,; by subtracting an appropriate multiple law of I from any other form UIA1 + .. · +u,,~, the last coefficient ~n can be eliminated. The linear forms of min u 1 ; • • • , Un -l which then remain form a submodule, and by the induction hypothesis this sub-

module has a linearly independent basis (11' ... , 1m _1) with m - 1 ~ n - I. Clearly

mis generated by 11' · .. , 1m -1, I. Now 11' ••• , Im-l are already linearly independent. If there were a linear dependence 11P1 + ... +Z... -tPm-t +IP = 0

with P =t= 0, then on equating coefficients of u" it would follow that /L,,{J = 0, which is impossible..

Exercises

IfIDl is a module of integral linear forms and if the submodule mis generated by finitely many linear forms v" = L Ujt%ib then a basis (It, ... ,1m) with the properties above can be constructed in a finite number of steps.

Using the basis (lh ... ,1m) constructed in Exercise 12.1, give a method of

determining whether a particular linear form Utfjt +. ·. + u,fln is contained

in the module 9l, that is, whether the linear system of diophantine

equations L ocik~k = fji is solvable in terms of integers el •

Modules Over Euclidean Rings. Elementary Divisors 5

If now the first row is added to the ith row using (2) and the first column multi-

plied by fJ is subtracted from the kth column by means of (3), then the element y

with g(y) < g«(Xi1) appears at the place (ik); this contradicts the minimality of (XII.

Our matrix now has the appearance

tXll 0 ... 0 o

A'

o

where all elements in A' are mUltiples of tXll. Using operations which leave the

first row and column unaltered, we now proceed with A' as previously with A.

The divisibility of all elements by tXll is hereby not destroyed. Finally A' acquires

the form

(X22 0 ... 0

o

A"

o

where all elements of A" are divisible by CX22. Continuing in this manner, we

obtain after m steps the desired normal form (12.1). It is not possible that one of

the matrices A, A', A", ... should consist solely of zeros before the form (12.7)

is obtained, since this would imply that certain of the Vk were equal to zero;

on the contrary, at each stage of the process the v form a linearly independent

basis of 91. This completes the proof of the theorem.

Remark 1: Operations 1 through 3 amount to multiplying the matrix A on

the right or left by an invertible matrix with elements in m. Indeed, if (Ui', ••• , u" ') = (Ui' · •• , U,,)· B and (VI', ••. , Vm ') = (VI' .•. , Vm )· C are new bases, then

(Vi' • • • Vm ') = (VI' • • V,JC = (Ul ... u,,)AC = (Ul' ... u,,')B- 1 AC.

The theorem on elementary divisors is therefore equivalent to the existence of two

invertible matrices Band C such that B- 1 A C is a matrix having the form (12.7).

Remark 2: The reduction of the matrix A proceeds in precisely the same

manner even if the V do not form a linearly independent system. In this case,

however, one of the matrices A, A', A" may become the zero matrix, and we

obtain instead of the normal form (12.7) the more general form

el 0

6 LUNEAR ALGEBRA

where r is the rank of A. The divisibility relations of the Bi remain the same.

Remark 3: The k-rowed subdeterminants of the transformed matrix D =

B- 1 AC are linear functions of the subdeterminants of A; similarly, those of

A = BDC- 1 are linear functions of the subdeterminants of D. Hence, up to

units the greatest common divisor 8 k of the k-rowed subdeterminants of A is

the same as for D. We easily obtain for D the value

(k~r).

Therefore

(1 <k~r). (12.9)

The 8 k are called the determinant divisors of the matrix A. The ek are called the

elementary divisors of A.1 From (12.9) it now follows that the elementary divisors

are the quotients o/two successive determinant divisors.

Remark 4: In the next section we shall find in another way that the elementary

divisors are uniquely determined up to units by the matrix A. It will be shown

that the elementary divisors (insofar as they are not units) depend only on the

factor module SJR/5Jl., which is in turn determined by A.

Exercise·

12.3 Any linear system of diophantine equations

(i = 1, ... , m)

with integers «'ik and fj, can be brought to the form

(i = 1, ... ,r; s, =1= 0)

(j = r + 1, ... , m)

by unimodular transformation of the unknowns and equations. The

conditions for the solvability of the system in terms of integers are

Yi =O(eJ; (^8) j = O. The '1, with i~r can be determined; the other 'YJj are arbitrary. The ek

are linear integral functions of the arbitrary 'r} j.'.

12.3 THE FUNDAMENTAL THEOREM OF ABELIAN GROUPS

Let (fj be an Abelian group with finitely many generators; (fj is a module, and

composition will be written additively. Ifm has an operator domain 91, then we

lThey are often called invariant factors in the English literature (Trans.).

8 LINEAR AWEBRA

We now proceed to the general case in which (fj is an 9t-module with finitely many generators gl' .. , , gIl; the elements of mthen have the form

'\lg1 + · ·. +',.g.. ,

If we form the module of linear forms

Wt = (u 1, ' • • , Uri)

with indeterminates U1' • • • , Un' then to each element L Aig (^) i of (fj there corre- sponds a linear form L AjUi ofIDl; the correspondence is again a module homo-

morphism, and it follows from the homomorphism theorem that

03 ~ IDllm, where 9l is the submodule of those linear forms L Aiu, for which L ~igi = O.

We again assume that 9t is Euclidean. By section 12.2 we can choose new

bases (V1' ... ,v~ and (U1', .•. ,u.. ') (n~m) for 91 and IDl such that

for i = 1, .. , , m

e, + 1 = O(e,).

The u' again correspond (under the homomorphism above) to elements hh. 0 • , hIt

of (Do All elements 'of (fj have the form JL1hl + · · · + ",,/t,,, and such an element is

zero if and only if

that is, ,..,.",+ 1 = 0

,..,.,. = o.

A sum ""'1 hi + · · · + ,..,.,.h,. is thus zero only if its individual terms are zero, and these are zero only if their coefficients il', are divisible by £, for i = 1, ..• , m and are zero for i = m+ 1, ... ,n.

This may be expressed as the following theorem.

Theorem: The group ffi is the direct sum of cyclic groups (h1) + 0 • • +(hll ), and' the annihilating ideal of (hi) is

(e i) for i = 1, •.• , m

(O) for i = m + 1, ... , n. This is the Fundamental Theorem of Abelian Groups with Finitely Many Generators. In the case of ordinary Abelian groups the leil are the orders of the cyclic

groups (hi)' ... , (h",), and the other groups (h", + 1)' ... , (hll ) have infinite order.

Three supplements to the theorem are still required:.

1. Elimination of units among the e i'
2. Further decomposition of the cyclic groups into prime-power groups.
3. Uniqueness.

The Fundamental Theorem of Abelian Groups 9

1. Suppose that 81, for instance, is a unit; (81) is then the unit ideal 9t and

hence 9th 1 = (0). The cyclic group 9th 1 may therefore be omitted from the

direct-sum decomposition 9thi + · · · + 9th•. Let the annihilating ideals (~i)' (0) which remain after elimination of units be written in reverse order as 01' ..• , at; then

al = O(a, + 1)·

2. Those groups (h.) whose annihilator is (0) are isomorphic to 91. Those

groups whose annihilator (el) =t= (0) can be further spilt into prime-power groups, as was demonstrated above. The annihilating prime powers are found by factoring

the ej. The sum of all the groups in the decomposition of ffi belonging to a prime

number p form a group ~"consisting of those elements ofm which are annihilated by a sufficiently high power ]II. Hence the groups ~" are uniquely determined. If U denotes the sum of the groups with.4 = (0), then we haye

(fj = L ~j,+U. p

By further decomposition of the ~p the prime-power groups are again obtained;

these are determined uniquely up to isomorphism, as we shall soon see. In each

$" there is a uniquely determined sequence of subgroups ~",Il' ~,,, -1, • • · '~P90'

where ~"." consists of those elements of ~p which are annihilated by p". The

first group of this sequence is is,; the last is the zero group.

The group U is uniquely determined up to isomorphism, since

U ~ (fj/L , ~".

3. Uniqueness Theorem: The annihilating ideals (11' •• • , Qq with Qt =O(Qi+1) for a direct sum decomposition <» = (tl + · · · +(tq are uniquely determined by

the module (fj alone. (Or equivalently: the groups G:i are uniquely determined up

to isomorphism.)

Proof: The asserted uniqueness is established once we have shown that it can

be uniquely determined how many of the ideals Qi are divisible by each prime

power pfl of~. Indeed, if pfl divides precisely k of these ideals, then these must

be the first k ideals 41' ... , 4t because of the divisibility property of these ideals. For every prime power ptl we thus know not only how many but also which

ideals are divisible by p"; this means that for each QI we know which prime

powers divide it. Those ideals which are divisible by arbitrarily high powers are zero; the others are uniquely determined by their prime factorization. If pfl divides the annihilator of the cyclic group [" then

ptl-l(f,,/p,,(t,

is a cyclic group with annihilator (p); it is therefore a simple group. On the other hand, if p(l does not divide the annihilating ideal of <t (^) l, then pfl<ti = p(l-l<t, and therefore p(l -1(f,i/pfl(f,j = (0). It follows that ptl-l(fj/ptl(f; is the direct sum of the