Table Of ContentOTHER TITLES IN THE SERIES ON
PURE AND APPLIED MATHEMATICS
Vol. 1. WALLACE—Introduction to Algebraic Topology
Vol. 2. PEDOE—Circles
Vol. 3. SPAIN—Analytical Conies
Vol. 4. MIKHLIN—Integral Equations
Vol. 5. EGGLESTON—Problems in Euclidean Space: Application of Convexity
Vol. 6. WALLACE—Homo logy Theory on Algebraic Varieties
Vol. 7. NOBLE—Methods Based on the Wiener-Hopf Technique for the Solution
of Partial Differential Equations
Vol. 8. MIKUSINSKI—Operational Calculus
Vol. 9. HEINE—Group Theory in Quantum Mechanics
Vol. 10. BLAND—The Theory of Linear Viscoelasticity
Vol. 11. KURTH—Axiomatics of Statistical Mechanics
Vol. 12. FUCHS—Abelian Groups
Vol. 13. KURATOWSKI—Introduction to Set Theory and Topology
Vol. 14. SPAIN—Analytical Quadrics
Vol. 15. HARTMAN and MIKUSINSKI—Theory of Measure and Lebesgue
Integration
Vol. 16. KULCZYCKI—Non-Euclidean Geometry
Vol. 17. KURATOWSKI—Introduction to Calculus
Vol. 18. GERONIMUS—Polynomials Orthogonal on a Circle and Interval
Vol. 19. ELSGOLC—Calculus of Variations
Vol. 20. ALEXITS—Convergence Problems of Orthogonal Series
Vol. 21. FUCHS and LEVIN—Functions of a Complex Variable Volume II
Vol. 22. GOODSTEIN—Fundamental Concepts of Mathematics
Vol. 23. KEENE—Abstract Sets and Finite Originals
Vol. 24. DITKIN and PRUDNIKOV—Operational Calculus in Two Variables and
its Applications
Vol. 25. VEKUA—Generalized Analytic Functions
Vol. 26. AMIR-MOEZ and FASS—Elements of Linear Spaces
Vol. 27. GRADSHTIEN—Direct and Converse Theorems
Vol. 28. FUCHS—Partially Ordered Algebraic Systems
M.M. POSTNIKOV
FOUNDATIONS
OF
GALOIS THEORY
translated by
ANN SWINFEN, B.A.(Oxon)
with
introductory foreword by
translation editor
P. J. HILTON
M.A., D.Phil. (Oxon), Ph.D (Cantab)
Mason Professor of Pure Mathematics
in the University of Birmingham
PERGAMON PRESS
OXFORD · LONDON · NEW YORK · PARIS
1962
PERGAMON PRESS LTD.
Headington Hill Hall, Oxford
4 & 5 Fitzroy Square, London, W.\
PERGAMON PRESS INC.
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PERGAMON PRESS G.m.b.H.
Kaiserstrasse 75, Frankfurt am Main
GAUTHER-VILLARS
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Copyright
©
1962
PERGAMON PRESS LTD.
Distributed in the Western Hemisphere by
THE MACMILLAN COMPANY · NEW YORK
pursuant to a special agreement with
PERGAMON PRESS LIMITED
Oxford, England
This translation has been made from Professor
M.M. Postnikov's book entitled "OSNOVY
TEORII GALUA" published in Moscow 1960
by Fizmatgiz
Library of Congress Card Number 62-19273
Set in Times No. 327, 10 on 12 pt.
by SANTYPE LIMITED, SALISBURY
and printed in Great Britain by THE PITMAN PRESS, BATH
FOREWORD
THE GALOIS theory of equations is simultaneously the source of
modern abstract algebra and one of the most concrete applications of
abstract algebra. For the problem of solving equations becomes
familiar at a very early stage of a school course in mathematics; and
the question, answered in the negative by Galois, whether all poly
nomial equations with integer coefficients can be solved by processes
involving only the ordinary operations of arithmetic together with
the extraction of nth roots must seem a natural and unavoidable one
to anyone well versed in school algebra.
Thus Galois theory constitutes an excellent introduction to group
theory and provides a strong (and historically significant) motivation
for the introduction of the basic concepts of abstract algebra. The
present work, by a leading Russian mathematician, sets itself the
limited aim of explaining the basic theory and describing how it is
applied to prove the unsolvability 'by radicals' of equations of
degree n ^ 5. On the other hand the treatment is almost entirely
self-contained (the necessary group theory, in particular, is given in
its entirety), and it is therefore hoped that this book will commend
itself to university students at the stage at which they first begin to
grapple with modern algebra.
Birmingham P. J. HILTON
vii
PREFACE
THIS BOOK is intended in the first instance for students in their second
or third year at university, who are starting to learn Galois theory.
For this reason the reader is assumed to possess only the fundamentals
of higher algebra to the extent of the programme of a first year course
at university. On the other hand, the theoretical material presented
in the book is not accompanied by examples, as it is assumed that
these will be provided in the course of lectures or seminars. The
problems included in the text are of a quite trivial character and are
intended solely for the self-discipline of the reader. It should be
noted that the order of presentation adopted in this book differs from
the order in which Galois theory should be presented in lectures (for
instance, groups of permutations should appear considerably earlier
in a course of lectures).
Galois theory is presented in this book for fields contained in some
unique 'universal', algebraically closed field of characteristic 0
(more specifically, the field of complex numbers). This makes it
possible to avoid the difficulty of founding an abstract theorem on the
existence and uniqueness (to within isomorphism) of the decomposi
tion field of a given polynomial. On the other hand, no actual loss
of generality occurs in such a method of presentation, because, as is
well known, any field can be embedded in an algebraically closed
field.
Another, less essential, peculiarity of the method of presentation
adopted in this book lies in the fact that we carefully avoid the use of
the theorem on the extension of an isomorphism, replacing it by
possibly less elegant, but certainly more accessible results in the
theory of symmetric functions. Further, we investigate, more
pedantically than is usually done, the relations between the different
definitions of a finite extension, and we base the presentation of the
group-theoretical material on the concept of a homomorphism (we
note in passing that for homomorphisms 'onto' and isomorphisms
'into' we use special terms, which have recently appeared in
mathematical literature and are rapidly superseding the customary
terminology).
viii
PREFACE
Because the theory of groups plays only a subsidiary part in Galois
theory, it is presented only in so far as it is necessary for the develop
ment of general Galois theory and its application to the problem of
solving equations by radicals. For instance, although we present
the concepts of a normal series and its refinement, yet there are no
theorems in this book like Schreier's theorem or the Jordan-Holder
theorem.
In presenting the theory of permutations, the theorem on the
factorization of permutations into the product of independent cycles
is proved in detail, and the concept of an even permutation emerges
from the consideration of the factorization of a permutation into
the product of transpositions. Although we do not insist on the
superiority of this method of introducing the concept of an even
permutation (as compared with the standard method, based on the
consideration of inversion in permutations), we nevertheless consider
that this method is worth attention. We prove the simplicity of the
alternating group as it was proved by Redei in a recent paper. Redei's
proof, as given by us, is simpler than the usual proof of Bauer.
In considering the solution of equations by radicals, we restrict
ourselves to the problem of the solution of equations by arbitrary
(possibly reducible) radicals. Thus the cyclotomic equations are
solvable by radicals by definition, and this, of course, essentially
simplifies the theory. Although with such an approach to the solu
tion of equations by radicals the results obtained cannot, for instance,
be applied to the problem of constructing regular polygons with the
help of ruler and compasses (as the theory of Gaussian periods
remains entirely outside the framework of our exposition), neverthe
less the simplification of the theory achieved in this way is so con
siderable that, for a preliminary introduction to the fundamental
ideas on which the application of Galois theory to the problem of
solution by radicals is based, the consideration of only irreducible
radicals is not appropriate.
In the last chapter of the book general equations (i.e. those
having literal coefficients) are considered. Because the field of co
efficients of these equations is the field of rational functions, if we
continue to treat the question from the point of view indicated above,
we are obliged in particular to prove that this field can be embedded
in an algebraically closed field (namely, in the field of fractional power
series). The fact that the field of fractional powers is algebraically
closed has been proved by Ostrowsky with the help of Hensel's
IX
PREFACE
lemma. This proof, although not effective, is considerably simpler
than the constructive proof, based on Newton's polygon and fre
quently reproduced in Russian.
As a reference book for the material of a preliminary course we use
A. G. Kurosh's book A Course in Higher Algebra, which in the text
is called simply 'The Course'. The page numbers given here are
those of the fourth and fifth editions.
The author takes this opportunity to thank V. G. Boltyanskii and
D. K. Faddeyev, who read this book in manuscript and made many
valuable suggestions. THE AUTHOR
x
CHAPTER 1
THE ELEMENTS OF FIELD THEORY
1. Preliminary remarks
We define a field as a non-empty set P of complex numbers,
possessing the following properties :
(1) if a e P and b e P, then a + beP and ab e P;
(2) if a e P, then -a e P and a'1 e P (when a φ 0).
Examples of fields are: the field of rational numbers R, the field
of real numbers D and the field of complex numbers C.
The field P is called a subfield of the field K, and the field K an
extension of the field P, if every element of the field P belongs to the
field K, i.e. iff P c K. Any field (in our sense) is a subfield of the
field of complex numbers.
It is easy to see that every field contains unity%, and hence also
the whole field of rational numbers R, i.e. any field is an extension of
the field of rational numbers.
In modern algebra one takes the abstract definition of a field as a
set with two algebraic operations satisfying definite axioms (see The
Course, p. 28). As opposed to such " abstract " fields, a field in our
sense is called a number field. The theory expounded in this book
can without difficulty also be extended to the case of arbitrary fields.
The transition from number fields to arbitrary fields involves, funda
mentally, only purely technical difficulties. These difficulties are
connected with the fact that in an arbitrary field some multiple of
unity can be equal to zero, and an irreducible polynomial can possess
multiple roots. Fields in which this difficulty does not arise are called
fields of characteristic 0 (see The Course, pp. 32 and 213). Besides
the number fields, the field of rational functions, for instance, belongs
to this class. Another more essential difficulty, arising in the transi
tion from number to arbitrary fields, lies, in particular, in the fact
that different fields, generally speaking, are in no way connected
t The notation PczK does not exclude the case when P coincides with K.
t [Editor's footnote: This does not follow from (1) and (2); one needs to
postulate that P contains at least two elements.]
1
2 FOUNDATIONS OF GALOIS THEORY
with one another: for instance, it is impossible to speak of the sum
of elements of two different fields. It is more convenient from every
point of view to overcome this difficulty by restricting the class of
fields considered to subfields of some sufficiently wide 'universal'
field. It is indeed in this way, by choosing as the universal field the
field of complex numbers, that we again arrive at the number fields.
In the general case it is sufficient to require that the universal field be
algebraically closed, i.e. to require that any polynomial over this
field is decomposable in it into linear factors. It is easy to verify that
all the theory expounded below remains valid without any alteration,
if by a field one understands a subfield of some fixed, but otherwise
arbitrary algebraically closed field of characteristic 0.
2. Some important types of extensions
An extension K of a field P is called finite, if in the field K there
exist elements oc ... , α„, such that any element ß e Ä^can be written
i9
in a unique way as a linear combination of these elements with
coefficients from the field P:
ß = fc^i + ... + b cc , b ... , b eP.
n n u n
A system of elements a ... , cc possessing this property is called a
1? n
basis of the field K over the field P.
One can also approach the concept of a finite extension from
another angle, by noting that any extension K of a field P can be
considered as a linear space over the field P. In fact, the elements of
the field K can be added together and multiplied by elements of the
field P, and the two operations (addition, and multiplication by
elements of the field P) clearly possess all the necessary properties.
From this point of view, the extension K is finite if and only if it has
finite dimension (as a linear space over the field P), and a system of
elements is a basis for it (in the sense defined) if and only if it is a
basis for it in the sense of the theory of linear spaces. Because all
bases of a finite-dimensional linear space have the same number of
vectors, then, in particular, all bases of the field K over the field P
have the same number of elements. This number is called the degree
of the field K over the field P and is denoted by [K : P] (from the point
of view of the theory of linear spaces, the degree of the field K is its
dimension as a linear space over the field P).
ELEMENTS OF FIELD THEORY 3
Problem. To prove that the degree [K : P] is equal to unity if and
only if K = P.
Let P be an arbitrary (number) field and a ... , a arbitrary
l5 n
numbers (i.e. elements of the field C). We consider all possible fields
which are extensions of the field P and contain the numbers ct ... , α .
u π
Such fields exist, since, for example, we may take the field C of all
complex numbers. It is easy to see that the intersection of all these
fields is also a field (in general, it is proved without difficulty that the
intersection of any family of fields is itself a field). This intersection
is, clearly, the minimal extension of the field P containing the
numbers a ... , a„ (the fact that it is minimal means that this inter
1?
section is a subfield of any other extension of P containing the num
bers a ... , a„). This minimal extension is denoted by P(a ... , a„)
1? l3
and is called the extension generated by the numbers a ... , oc .
l5 n
It is clear that P(a ... , a„) = P if and only if a ... , a„ e P.
l5 l5
Problem. To prove that the field P{a ... , α„) can be defined as
u
the set of all numbers obtained as the result of applying to the
numbers of the field P and the numbers a ... , a all possible
l5 M
combinations of the four arithmetical operations.
The number a is called algebraic over the field P, if it is the root of
some polynomial with coefficients from the field P (not identically
equal to zero). Any element of the field P, clearly, is algebraic over
this field (if the converse is also true, i.e. if any number algebraic
over the field P belongs to this field, then P is called an algebraically
closed field; see para. 1). It is clear, moreover, that any number
algebraic over the field P is also algebraic over any extension of
the field P. We emphasize that the converse assertion, generally
speaking, is not true. For instance, any complex number is algebraic
over the field D of real numbers (because it is the root of a quadratic
trinomial with real coefficients), whilst there exist numbers (indeed,
real numbers), not algebraic over the field R of rational num
bers. As an example of numbers non-algebraic over the field R
one can mention the well-known numbers e and π, whose non-
algebraic nature is proved in complete courses on the theory of
numbers.
An extension K of the field P is called algebraically generated if it
is generated by some finite system of numbers algebraic over the
field JP, i.e. if there exist numbers oq, ... , a algebraic over the field P
s
such that K = P(OLU ... , as). If, in particular, 5=1, then the field
K = Picti) is called a simple algebraic extension of the field P.
