Topics in Field Theory

559 Pages·1989·16.756 MB·English

by  Gregory Karpilovsky (Eds.)

TOPICS IN FIELD THEORY
NORTH-HOLLAND MATHEMATICS STUDIES 155
Notas de Matematica (12 4)
Editor: Leopoldo Nachbin
Centro Brasileiro de Pesquisas Fisicas
Rio de Janeiro
and University of Rochester
NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD *TOKM
TOPICS IN FIELD THEORY
G rego ry KARP IL OVS KY
Department of Mathematics
University of the Witwatersrand
Johannesburg, South Africa
1989
NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD *TOKYO
' Elsevier Science Publishers B.V., 1989
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Library of Congress Cataloging-in-Publication Data
Karpilovsky. Gregory, 1940-
Topics in field theory / Gregory Karpilovsky.
p. cm. -- (North-Holland mathematics studles ; 155) (Notas
oe maternitica ; 124)
Bibliography: p.
Includes index.
ISBN 0-444-87297-3
1. Fields, Aljsbraic. I. Title. 11. Serles. 111. Series: Notas
.
de matembtica (Rio de Ja?eiro. Brazil) no. 124.
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vi i
Preface
The present book is intended to give a systematic account of certain important topics
pertaining to field theory. The author has tried to be fairly complete in what he considers
as the main body of the theory and the reader should get a considerable amount of
knowledge of central ideas, the basic results, and the fundamental methods. We have tried
to avoid making the discussion too technical. With this view in mind, maximum generality
has not been achieved in those places where this would entail a loss of clarity or a lot of
technicalities.
The present monograph is written on the assumption that the reader has had the
equivalent of a standard first-year graduate algebra course. Thus we assume a familiarity
with basic ring-theoretic and group-theoretic concepts. For the convenience of the reader,
a chapter on algebraic preliminaries is included. There is a fairly large bibliography of
works which are either directly relevant to the text or offer supplementary material of
interest. The following is a brief description of the content of the book.
After establishing algebraic preliminaries (Chapter l), we concentrate on separable
algebraic extensions (Chapter 2). Among other results, we provide a characterization of
finite separable field extensions E/F in terms of the number of F-homomorphisms of E
into its algebraic closure. We then characterize separability via linear disjointness and
tensor products. Turning to the trace map and discriminants, we characterize separability
via these notions.
Chapter 3 is confined to a systematic study of transcendental extensions. We begin
by introducing abstract dependence relations, which will allow us to treat in a unified
manner algebraic independence and p-independence. Special attention is drawn to the
investigation of the existence of separating transcendency basis. Extensions E/F with
separating transcendency bases are special instances of extensions E/F which preserve p-
independence. The latter extensions are characterized in a large number of different ways.
The chapter culminates in the study of relatively separated and reliable extensions.
viii PREFACE
In Chapter 4 we introduce derivations of fields and present a number of their important
properties. Among these, we provide various criteria for extending the derivations and
characterize separability via derivations. Given a field E of characteristic p > 0, we also
exhibit a bijection between the set of subfields of E containing EP and the set of closed
restricted subspaces of DerE.
Chapter 5 is devoted to a detailed study of purely inseparable extensions. Our pre-
sentation of the theory of modular purely inseparable extensions is based on an important
work of Waterhouse (1975). The basic discovery of Waterhouse is that the theory is closely
related to the well developed study of primary abelian groups. After establishing some
preliminary results, we develop the theory of pure independence, basic subfields, and ten-
sor products of simple extensions. We then compute the Ulm invariants and display some
complications in the field extensions not occurring in abelian groups. The final section is
devoted to modular closure and modularly perfect fields.
In Chapter 6 we piesent the Galois theory which may be described as the analysis
of field extensions by means of automorphism groups Special attention is drawn to the
problem of realizing finite groups as Galois groups. In particular, we show that certain
types of split extensions of elementary abelian 2-group by the realizable group G occur
as Galois groups of normal real extensions. The chapter ends with a brief discussion of
Galois cohomology.
Chapter 7 is devoted to the study of abelian extensions, i.e. Galois extensions with
abelian Galois groups. Among other results, we provide an explicit description of all cyclic
extensions of degree pn, p prime, of a given field F containing all pn-th roots of unity.
We then study abelian pextensions by means of Witt vectors. After presenting Kummer
theory, we finally exhibit a bijective correspondence between the subgroups of the character
group of Gal(E/F) (E/F is a Galois extension) and all abelian subextensions of E/F. This
is achieved by applying infinite Galois theory and certain properties of character groups
of prohite groups.
Chapter 8, the final chapter is devoted to a detailed investigation of radical extensions.
1 would like to express my gratitude to my wife for the encoiiragement she has given
me in the preparation of this book. Finally, my thanks go to Lucy Rich for her excellent
typing.
ix
Contents
PREFACE vii
CHAPTER 1. ALGEBRAIC PRELIMINARIES 1
1. Notation and terminology 1
2. Localization 6
3. Integral extensions 9
4. Polynomial rings 14
5. Unique factorization domains 24
6. Dedekind domains 32
CHAPTER 2. SEPARABLE ALGEBRAIC EXTENSIONS 45
1. Algebraic closure, splitting fields and normal extensions 45
2. Separable algebraic extensions: definitions and elementary properties 58
3. Separability, linear disjointness and tensor products 73
4. Norms, traces and discriminants of separable field extensions 82
CHAPTER 3. TRANSCENDENTAL EXTENSIONS 97
1. Abstract dependence relations 97
2. Transcendency bases 100
3. Simple transcendental extensions 106
4. Separable extensions 109
5. Weil’s order of inseparability 125
6. Separability and preservation of pindependence 131
7. Perfect ground fields 142
8. Criteria for separating transcendency bases 147
9. Separable generation of intermediate field extensions 153
10. The Steinitz field tower 156
11. Nonseparably generated fields over maximal perfect subfields 167
12. Relatively separated extensions 172
13. Reliability and relative separability 181