Algebraic Numbers and Algebraic Functions

205 Pages·2017·8.53 MB·English

by  Cohn

Algebraic Numbers and
Algebraic Functions

Algebraic Numbers and
Algebraic Functions
P. M. Cohn, FRS
Department of Mathematics
University College London
CRC Press
Taylor & Francis Group
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
First published 1991 by CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
Reissued 2018 by CRC Press
© 1991 by P. M. Cohn
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have
been made to publish reliable data and information, but the author and publisher cannot assume responsibility for
the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the
copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to
publish in this form has not been obtained. If any copyright material has not been acknowledged please write and
let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or
utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including
photocopying, microfilming, and recording, or in any information storage or retrieval system, without written
permission from the publishers.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication data available
A Library of Congress record exists under LC control number: 92131337
Publisher’s Note
The publisher has gone to great lengths to ensure the quality of this reprint but points out that some imperfections
in the original copies may be apparent.
Disclaimer
The publisher has made every effort to trace copyright holders and welcomes correspondence from those they
have been unable to contact.
ISBN 13: 978-1-315-89048-7 (hbk)
ISBN 13: 978-1-351-06958-8 (ebk)
Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the
CRC Press Web site at http://www.crcpress.com
Contents
Preface vii
Notes to the reader ix
Introduction xi
1(cid:9) Fields with valuations
1.1(cid:9) Absolute values 1
1.2(cid:9) The topology defined by an absolute value 6
1.3(cid:9) Complete fields 11
1.4(cid:9) Valuations, valuation rings and places 18
1.5(cid:9) The representation by power series 27
1.6(cid:9) Ordered groups 31
1.7(cid:9) General valuations 37
2 Extensions
2.1 Generalities on extensions 43
2.2 Extensions of complete fields 46
2.3 Extensions of incomplete fields 54
2.4 Dedekind domains and the strong approximation theorem 58
2.5 Extensions of Dedekind domains 67
2.6 Different and discriminant 72
3 Global fields
3.1 Algebraic number fields 83
3.2 The product formula 91
3.3 The unit theorem 100
3.4 The class number 106
4 Function fields
4.1 Divisors on a function field 109
4.2 Principal divisors and the divisor class group 118
4.3 Riemann's theorem and the specialty index 125
vi Contents
4.4 The genus(cid:9) 129
4.5 Derivations and differentials(cid:9) 135
4.6 The Riemann—Roch theorem and its consequences(cid:9) 144
4.7 Elliptic function fields(cid:9) 155
4.8 Abelian integrals and the Abel—Jacobi theorem(cid:9) 167
5 Algebraic function fields in two variables
5.1(cid:9) Valuations on function fields of two variables(cid:9) 176
Bibliography(cid:9) 183
Table of notations(cid:9) 187
Index (cid:9) 189
Preface
One of the most interesting and central areas of mathematics is the theory of
algebraic functions — algebra, analysis and geometry meet here and interact in
a significant way. Some years ago I gave a course on this topic in the Univer-
sity of London, using Artin's approach via valuations, which allowed one to
treat algebraic numbers and functions in parallel. At the invitation of my
friend Paulo Ribenboim I prepared a set of lecture notes which was issued by
Queen's University, Kingston, Ontario, but I had always felt that they might
be deserving of a wider audience.
Ideally one would first develop the algebraic, analytic and geometric back-
ground and then pursue the theme along these three paths of its development.
This would have resulted in a massive and not very readable tome. Instead I
decided to assume the necessary background from algebra and complex anal-
ysis and leave out the geometric aspects, except for the occasional aside, but
to give a fairly full exposition of the necessary valuation theory. This allowed
the text to be kept to a reasonable size.
Chapter 1 is an account of valuation theory, including all that is needed
later, and it could be read independently as a concise introduction to a power-
ful method of studying general fields. Chapter 2 describes the behaviour
under extensions and shows how Dedekind domains can be characterized as
rings of integers for a family of valuations with the strong approximation
property; this makes the passage to extensions particularly transparent. These
methods are then put to use in Chapter 3 to characterize global fields by
means of the product formula, to classify the global fields and to prove two
basic results of algebraic number theory in this context: the unit theorem and
the finiteness of the class number.
Chapter 4, the longest in the book, treats algebraic function fields of one
variable. The main aim is the description of the group of divisor classes of
degree zero as a particular 2g-dimensional compact abelian group, the
Jacobian variety. This is the Abel—Jacobi theorem; the methods used are as far
as possible algebraic, although the function—theoretic interpretation is borne
viii Preface
in mind. On the way automorphisms of function fields are discussed and the
special case of elliptic function fields is developed in a little more detail. A
final brief chapter examines the case of valuations on fields of two variables;
in a sense this is a continuation of Chapter 1 which it is hoped will illuminate
the development of Chapter 4. The main exposition follows the notes, but the
telegraphese style has been expanded and clarified where necessary, and there
have been several substantial additions, especially in Chapter 4.
The prerequisites are quite small: an undergraduate course in algebra and in
complex function theory should suffice; references are generally given for any
results needed. There are a number of exercises containing examples and
further developments.
In writing a book of this kind one inevitably has a heavy debt to others.
The first three chapters were much influenced by the writings of E. Artin
quoted in the bibliography. A course on number theory by J. Dieudonne
(which I attended at an impressionable age) has helped me greatly in Chapter
2. The treatment of valuations on function fields of two variables (Chapter
5) is based on a paper by Zariski, while Chapter 4 owes much to several
works in the bibliography, particularly those by Hensel and Landsberg,
Tchebotarev and by Eichler. I should like to thank Paulo Ribenboim for
bringing out the earlier set of notes and a number of friends and colleagues
for their advice and help, particularly David Eisenbud, Frank E. A. Johnson
and Mark L. Roberts. The latter has also helped with the proof reading.
Further I am indebted to the staff of Chapman & Hall for the way they have
accommodated my wishes.
P. M. Cohn
University College London
May 1991
Notes to the reader
This book is intended for readers who have a basic background in algebra
(elementary notions of groups, rings, fields and linear algebra). Occasionally
results from Galois theory or commutative ring theory are used, but in such
cases references are usually given, mostly to the author's algebra text,
referred to as A.1, 2, 3 (see the bibliography). For Chapter 4 an acquaintance
with the elements of complex function theory will be helpful.
Using notation which, thanks to Bourbaki, has become standard, we write
N, Z, Q, R, C for the natural numbers, integers, rational, real and complex
numbers respectively and Fp for the field of p elements (integers mod p).
Rings are always associative, with a unit element, written 1, and except for
some occasions in Chapter 1 they are commutative. If 1 = 0, the ring is re-
duced to 0 and is called trivial; mostly our rings are non-trivial. If K is any
ring, the polynomial ring in an indeterminate x over K is written K[x]; when
K is a field, K[x] has a field of fractions, called the field of rational functions
in x over K and written K(x). A polynomial is called monic if its leading
coefficient is 1. A ring R is called a K-algebra if it is a K-module and the
multiplication in R is K-linear in each argument, e.g. K[x] is a K-algebra. The
additive group of a ring R is denoted by R+ and the set of non-zero elements
in R by R"; if this set contains 1 and is closed under multiplication, R is called
an integral domain. In a group G the subgroup generated by a subset X is
denoted by (X). If H is a subgroup of G, a transversal of H in G is a complete
set of representatives (for the cosets of H in G). A set with an associative
binary operation is a semigroup; if it also contains a neutral element for the
multiplication (e such that ex = xe = x for all x) it is called a monoid.
In one or two places Zorn's lemma is used; we recall that this is the
algebraist's version of the axiom of choice. It states that a partially ordered
set A has a maximal element provided that every totally ordered subset has an
upper bound in A. If every totally ordered subset in A has an upper bound, A
is called inductive; so the lemma states that every inductive partially ordered
set has a maximal element (cf. A.2, section 1.2 for a discussion). A property