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Graduate Texts in Mathematics
254
Editorial Board
S. Axler K.A. Ribet
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Graduate Texts in Mathematics
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TAKEUTI/ZARING. Introduction to
Axiomatic Set Theory. 2nd ed.
OXTOBY. Measure and Category. 2nd ed.
SCHAEFER. Topological Vector Spaces.
2nd ed.
HILTON/STAMMBACH. A Course in
Homological Algebra. 2nd ed.
MAC LANE. Categories for the Working
Mathematician. 2nd ed.
HUGHES/PIPER. Projective Planes.
J.-P. SERRE. A Course in Arithmetic.
TAKEUTI/ZARING. Axiomatic Set Theory.
HUMPHREYS. Introduction to Lie Algebras
and Representation Theory.
COHEN. A Course in Simple Homotopy
Theory.
CONWAY. Functions of One Complex
Variable I. 2nd ed.
BEALS. Advanced Mathematical Analysis.
ANDERSON/FULLER. Rings and Categories of
Modules. 2nd ed.
GOLUBITSKY/GUILLEMIN. Stable Mappings
and Their Singularities.
BERBERIAN. Lectures in Functional Analysis
and Operator Theory.
WINTER. The Structure of Fields.
ROSENBLATT. Random Processes. 2nd ed.
HALMOS. Measure Theory.
HALMOS. A Hilbert Space Problem Book.
2nd ed.
HUSEMOLLER. Fibre Bundles. 3rd ed.
HUMPHREYS. Linear Algebraic Groups.
BARNES/MACK. An Algebraic Introduction
to Mathematical Logic.
GREUB. Linear Algebra. 4th ed.
HOLMES. Geometric Functional Analysis
and Its Applications.
HEWITT/STROMBERG. Real and Abstract
Analysis.
MANES. Algebraic Theories.
KELLEY. General Topology.
ZARISKI/SAMUEL. Commutative Algebra.
Vol.I.
ZARISKI/SAMUEL. Commutative Algebra.
Vol.II.
JACOBSON. Lectures in Abstract Algebra I.
Basic Concepts.
JACOBSON. Lectures in Abstract Algebra II.
Linear Algebra.
JACOBSON. Lectures in Abstract Algebra III.
Theory of Fields and Galois Theory.
HIRSCH. Differential Topology.
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SPITZER. Principles of Random Walk.
2nd ed.
ALEXANDER/WERMER. Several Complex
Variables and Banach Algebras. 3rd ed.
KELLEY/NAMIOKA et al. Linear
Topological Spaces.
MONK. Mathematical Logic.
GRAUERT/FRITZSCHE. Several Complex
Variables.
ARVESON. An Invitation to C*-Algebras.
KEMENY/SNELL/KNAPP. Denumerable
Markov Chains. 2nd ed.
APOSTOL. Modular Functions and
Dirichlet Series in Number Theory.
2nd ed.
J.-P. SERRE. Linear Representations of
Finite Groups.
GILLMAN/JERISON. Rings of Continuous
Functions.
KENDIG. Elementary Algebraic Geometry.
LOÈVE. Probability Theory I. 4th ed.
LOÈVE. Probability Theory II. 4th ed.
MOISE. Geometric Topology in
Dimensions 2 and 3.
SACHS/WU. General Relativity for
Mathematicians.
GRUENBERG/WEIR. Linear Geometry.
2nd ed.
EDWARDS. Fermat’s Last Theorem.
KLINGENBERG. A Course in Differential
Geometry.
HARTSHORNE. Algebraic Geometry.
MANIN. A Course in Mathematical Logic.
GRAVER/WATKINS. Combinatorics with
Emphasis on the Theory of Graphs.
BROWN/PEARCY. Introduction to Operator
Theory I: Elements of Functional Analysis.
MASSEY. Algebraic Topology: An
Introduction.
CROWELL/FOX. Introduction to Knot Theory.
KOBLITZ. p-adic Numbers, p-adic Analysis,
and Zeta-Functions. 2nd ed.
LANG. Cyclotomic Fields.
ARNOLD. Mathematical Methods in
Classical Mechanics. 2nd ed.
WHITEHEAD. Elements of Homotopy
Theory.
KARGAPOLOV/MERLZJAKOV. Fundamentals
of the Theory of Groups.
BOLLOBAS. Graph Theory.
EDWARDS. Fourier Series. Vol. I. 2nd ed.
WELLS. Differential Analysis on Complex
Manifolds. 2nd ed.
(continued after index)
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Henning Stichtenoth
Sabanci University
Faculty of Engineering & Natural Sciences
34956 Istanbul
Orhanli, Tuzla
Turkey
Editorial Board
S. Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132
USA
K.A. Ribet
Mathematics Department
University of California, Berkeley
Berkeley, CA 94720-3840
USA
ISSN: 0072-5285
ISBN: 978-3-540-76877-7
e-ISBN: 978-3-540-76878-4
Library of Congress Control Number: 2008938193
Mathematics Subject Classification (2000): 12xx, 94xx, 14xx, 11xx
c 2009 Springer-Verlag Berlin Heidelberg
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Preface to the Second Edition
15 years after the first printing of Algebraic Function Fields and Codes, the
mathematics editors of Springer Verlag encouraged me to revise and extend
the book.
Besides numerous minor corrections and amendments, the second edition
differs from the first one in two respects. Firstly I have included a series
of exercises at the end of each chapter. Some of these exercises are fairly
easy and should help the reader to understand the basic concepts, others are
more advanced and cover additional material. Secondly a new chapter titled
“Asymptotic Bounds for the Number of Rational Places” has been added.
This chapter contains a detailed presentation of the asymptotic theory of
function fields over finite fields, including the explicit construction of some
asymptotically good and optimal towers. Based on these towers, a complete
and self-contained proof of the Tsfasman-Vladut-Zink Theorem is given. This
theorem is perhaps the most beautiful application of function fields to coding
theory.
The codes which are constructed from algebraic function fields were first
introduced by V. D. Goppa. Accordingly I referred to them in the first edition
as geometric Goppa codes. Since this terminology has not generally been accepted in the literature, I now use the more common term algebraic geometry
codes or AG codes.
I would like to thank Alp Bassa, Arnaldo Garcia, Cem Gă
uneri, Sevan
Harput and Alev Topuzo
glu for their help in preparing the second edition.
Moreover I thank all those whose results I have used in the exercises without
giving references to their work.
˙
Istanbul,
September 2008
Henning Stichtenoth
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Preface to the First Edition
An algebraic function field over K is an algebraic extension of finite degree
over the rational function field K(x) (the ground field K may be an arbitrary
field). This type of field extension occurs naturally in various branches of
mathematics such as algebraic geometry, number theory and the theory of
compact Riemann surfaces. Hence one can study algebraic function fields from
very different points of view.
In algebraic geometry one is interested in the geometric properties of an
algebraic curve C = {(α, β) ∈ K × K | f (α, β) = 0}, where f (X, Y ) is an
irreducible polynomial in two variables over an algebraically closed field K.
It turns out that the field K(C) of rational functions on C (which is an
algebraic function field over K) contains a great deal of information regarding
the geometry of the curve C. This aspect of the theory of algebraic function
fields is presented in several books on algebraic geometry, for instance [11],
[18], [37] and [38].
One can also approach function fields from the direction of complex analysis. The meromorphic functions on a compact Riemann surface S form an
algebraic function field M(S) over the field C of complex numbers. Here again,
the function field is a strong tool for studying the corresponding Riemann surface, see [10] or [20].
In this book a self-contained, purely algebraic exposition of the theory
of algebraic functions is given. This approach was initiated by R. Dedekind,
L. Kronecker and H. M. Weber in the nineteenth century (over the field C), cf.
[20]; it was further developed by E. Artin, H. Hasse, F. K. Schmidt and A. Weil
in the first half of the twentieth century. Standard references are Chevalley’s
book ‘Introduction to the Theory of Algebraic Functions of One Variable’ [6],
which appeared in 1951, and [7]. The close relationship with algebraic number
theory is emphasized in [1] and [9].
The algebraic approach to algebraic functions is more elementary than the
approach via algebraic geometry: only some basic knowledge of algebraic field
extensions, including Galois theory, is assumed. A second advantage is that
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VIII
Preface to the First Edition
some principal results of the theory (such as the Riemann-Roch Theorem) can
be derived very quickly for function fields over an arbitrary constant field K.
This facilitates the presentation of some applications of algebraic functions to
coding theory, which is the second objective of the book.
An error-correcting code is a subspace of IFnq , the n-dimensional standard
vector space over a finite field IFq . Such codes are in widespread use for the
reliable transmission of information. As observed by V. D. Goppa in 1975, one
can use algebraic function fields over IFq to construct a large class of interesting codes. Properties of these codes are closely related to properties of the
corresponding function field, and the Riemann-Roch Theorem provides estimates, sharp in many cases, for their main parameters (dimension, minimum
distance).
While Goppa’s construction is the most important, it is not the only link
between codes and algebraic functions. For instance, the Hasse-Weil Theorem
(which is fundamental to the theory of function fields over a finite constant
field) yields results on the weights of codewords in certain trace codes.
A brief summary of the book follows.
The general theory of algebraic function fields is presented in Chapters 1,
3 and 4. In the first chapter the basic concepts are introduced, and A. Weil’s
proof of the Riemann-Roch Theorem is given. Chapter 3 is perhaps the most
important. It provides the tools necessary for working with concrete function fields: the decomposition of places in a finite extension, ramification and
different, the Hurwitz Genus Formula, and the theory of constant field extensions. P -adic completions as well as the relation between differentials and
Weil differentials are treated in Chapter 4.
Chapter 5 deals with function fields over a finite constant field. This chapter contains a version of Bombieri’s proof of the Hasse-Weil Theorem as well as
some improvements of the Hasse-Weil Bound. As an illustration of the general
theory, several explicit examples of function fields are discussed in Chapter 6,
namely elliptic and hyperelliptic function fields, Kummer and Artin-Schreier
extensions of the rational function field.
The Chapters 2, 8 and 9 are devoted to applications of algebraic functions
to coding theory. Following a brief introduction to coding theory, Goppa’s
construction of codes by means of an algebraic function field is described
in Chapter 2. Also included in this chapter is the relation these codes have
with the important classes of BCH and classical Goppa codes. Chapter 8 contains some supplements: the residue representation of geometric Goppa codes,
automorphisms of codes, asymptotic questions and the decoding of geometric Goppa codes. A detailed exposition of codes associated to the Hermitian
function field is given. In the literature these codes often serve as a test for
the usefulness of geometric Goppa codes.
Chapter 9 contains some results on subfield subcodes and trace codes.
Estimates for their dimension are given, and the Hasse-Weil Bound is used
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Preface to the First Edition
IX
to obtain results on the weights, dimension and minimum distance of these
codes.
For the convenience of the reader, two appendices are enclosed. Appendix
A is a summary of results from field theory that are frequently used in the
text. As many papers on geometric Goppa codes are written in the language
of algebraic geometry, Appendix B provides a kind of dictionary between the
theory of algebraic functions and the theory of algebraic curves.
Acknowledgements
First of all I am indebted to P. Roquette from whom I learnt the theory
of algebraic functions. His lectures, given 20 years ago at the University of
Heidelberg, substantially influenced my exposition of this theory.
I thank several colleagues who carefully read the manuscript: D. Ehrhard,
P. V. Kumar, J. P. Pedersen, H.-G. Ră
uck, C. Voss and K. Yang. They suggested many improvements and helped to eliminate numerous misprints and
minor mistakes in earlier versions.
Essen, March 1993
Henning Stichtenoth
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Contents
1
Foundations of the Theory of Algebraic Function Fields . . .
1.1 Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Rational Function Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Independence of Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 The Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Some Consequences of the Riemann-Roch Theorem . . . . . . . . . .
1.7 Local Components of Weil Differentials . . . . . . . . . . . . . . . . . . . . .
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
8
12
15
24
31
37
40
2
Algebraic Geometry Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 AG Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Rational AG Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
45
48
55
63
3
Extensions of Algebraic Function Fields . . . . . . . . . . . . . . . . . . . 67
3.1 Algebraic Extensions of Function Fields . . . . . . . . . . . . . . . . . . . . 68
3.2 Subrings of Function Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Local Integral Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4 The Cotrace of Weil Differentials and the Hurwitz Genus
Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5 The Different . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.6 Constant Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.7 Galois Extensions I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.8 Galois Extensions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.9 Ramification and Splitting in the Compositum of Function
Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.10 Inseparable Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.11 Estimates for the Genus of a Function Field . . . . . . . . . . . . . . . . 145
3.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
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Contents
4
Differentials of Algebraic Function Fields . . . . . . . . . . . . . . . . . . 155
4.1 Derivations and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.2 The P -adic Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4.3 Differentials and Weil Differentials . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5
Algebraic Function Fields over Finite Constant Fields . . . . . 185
5.1 The Zeta Function of a Function Field . . . . . . . . . . . . . . . . . . . . . 185
5.2 The Hasse-Weil Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.3 Improvements of the Hasse-Weil Bound . . . . . . . . . . . . . . . . . . . . 208
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6
Examples of Algebraic Function Fields . . . . . . . . . . . . . . . . . . . . . 217
6.1 Elliptic Function Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.2 Hyperelliptic Function Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.3 Tame Cyclic Extensions of the Rational Function Field . . . . . . . 227
6.4 Some Elementary Abelian p-Extensions of K(x),
char K = p > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
7
Asymptotic Bounds for the Number of Rational Places . . . . 243
7.1 Ihara’s Constant A(q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.2 Towers of Function Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.3 Some Tame Towers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.4 Some Wild Towers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
8
More about Algebraic Geometry Codes . . . . . . . . . . . . . . . . . . . . 289
8.1 The Residue Representation of CΩ (D, G) . . . . . . . . . . . . . . . . . . . 289
8.2 Automorphisms of AG Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
8.3 Hermitian Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
8.4 The Tsfasman-Vladut-Zink Theorem . . . . . . . . . . . . . . . . . . . . . . . 297
8.5 Decoding AG Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
9
Subfield Subcodes and Trace Codes . . . . . . . . . . . . . . . . . . . . . . . . 311
9.1 On the Dimension of Subfield Subcodes and Trace Codes . . . . . 311
9.2 Weights of Trace Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
Appendix A. Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Appendix B. Algebraic Curves and Function Fields . . . . . . . . . . . . 335
List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
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Contents
XIII
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
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1
Foundations of the Theory of Algebraic
Function Fields
In this chapter we introduce the basic definitions and results of the theory of
algebraic function fields: valuations, places, divisors, the genus of a function
field, adeles, Weil differentials and the Riemann-Roch Theorem.
Throughout Chapter 1 we denote by K an arbitrary field.
It is only in later chapters that we will assume that K has specific properties
(for example, that K is a finite field – the case which is of particular interest
to coding theory).
1.1 Places
Definition 1.1.1. An algebraic function field F/K of one variable over K is
an extension field F ⊇ K such that F is a finite algebraic extension of K(x)
for some element x ∈ F which is transcendental over K.
For brevity we shall simply refer to F/K as a function field. Obviously the set
˜ := {z ∈ F | z is algebraic over K} is a subfield of F , since sums, products
K
˜ is called the field
and inverses of algebraic elements are also algebraic. K
˜
˜ is a
of constants of F/K. We have K ⊆ K
F , and it is clear that F/K
˜
function field over K. We say that K is algebraically closed in F (or K is the
˜ = K.
full constant field of F ) if K
Remark 1.1.2. The elements of F which are transcendental over K can be
characterized as follows: z ∈ F is transcendental over K if and only if the
extension F/K(z) is of finite degree. The proof is trivial.
Example 1.1.3. The simplest example of an algebraic function field is the rational function field; F/K is called rational if F = K(x) for some x ∈ F
which is transcendental over K. Each element 0 = z ∈ K(x) has a unique
representation
H. Stichtenoth, Algebraic Function Fields and Codes,
Graduate Texts in Mathematics 254,
c Springer-Verlag Berlin Heidelberg 2009
1
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2
1 Foundations of the Theory of Algebraic Function Fields
z =a·
pi (x)ni ,
(1.1)
i
in which 0 = a ∈ K, the polynomials pi (x) ∈ K[x] are monic, pairwise distinct
and irreducible and ni ∈ ZZ.
A function field F/K is often represented as a simple algebraic field extension
of a rational function field K(x); i.e., F = K(x, y) where ϕ(y) = 0 for some irreducible polynomial ϕ(T ) ∈ K(x)[T ]. If F/K is a non-rational function field,
it is not so clear, whether every element 0 = z ∈ F admits a decomposition
into irreducibles analogous to (1.1); indeed, it is not even clear what we mean
by an irreducible element of F . Another problem which is closely related to
the representation (1.1) is the following: given elements α1 , . . . , αn ∈ K, find
all rational functions f (x) ∈ K(x) with a prescribed order of zero (or pole order) at α1 , . . . , αn . In order to formulate these problems for arbitrary function
fields properly, we introduce the notions of valuation rings and places.
Definition 1.1.4. A valuation ring of the function field F/K is a ring O ⊆ F
with the following properties:
(1) K O F , and
(2) for every z ∈ F we have that z ∈ O or z −1 ∈ O.
This definition is motivated by the following observation in the case of a
rational function field K(x): given an irreducible monic polynomial p(x) ∈
K[x], we consider the set
Op(x) :=
f (x)
f (x), g(x) ∈ K[x], p(x) g(x)
g(x)
.
It is easily verified that Op(x) is a valuation ring of K(x)/K. Note that if q(x)
is another irreducible monic polynomial, then Op(x) = Oq(x) .
Proposition 1.1.5. Let O be a valuation ring of the function field F/K. Then
the following hold:
(a) O is a local ring; i.e., O has a unique maximal ideal P = O \ O× , where
O× = {z ∈ O | there is an element w ∈ O with zw = 1} is the group of
units of O.
(b) Let 0 = x ∈ F . Then x ∈ P ⇐⇒ x−1 ∈ O.
˜ of constants of F/K we have K
˜ ⊆ O and K
˜ ∩ P = {0}.
(c) For the field K
Proof. (a) We claim that P := O \ O× is an ideal of O (from this it follows
at once that P is the unique maximal ideal since a proper ideal of O cannot
contain a unit).
(1) Let x ∈ P , z ∈ O . Then xz ∈ O× (otherwise x would be a unit),
consequently xz ∈ P .
(2) Let x, y ∈ P . W.l.o.g. we can assume that x/y ∈ O. Then 1 + x/y ∈ O
and x + y = y(1 + x/y) ∈ P by (1). Hence P is an ideal of O.
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1.1 Places
3
(b) is obvious.
˜ Assume that z ∈ O . Then z −1 ∈ O as O is a valuation
(c) Let z ∈ K.
−1
is algebraic over K, there are elements a1 , . . . , ar ∈ K with
ring. Since z
ar (z −1 )r + . . . + a1 z −1 + 1 = 0, hence z −1 (ar (z −1 )r−1 + . . . + a1 ) = −1.
Therefore z = −(ar (z −1 )r−1 + . . . + a1 ) ∈ K[z −1 ] ⊆ O, so z ∈ O. This is a
˜ ⊆ O.
contradiction to the assumption z ∈ O. Hence we have shown that K
˜ ∩ P = {0} is trivial.
The assertion K
Theorem 1.1.6. Let O be a valuation ring of the function field F/K and let
P be its unique maximal ideal. Then the following hold:
(a) P is a principal ideal.
(b) If P = tO then each 0 = z ∈ F has a unique representation of the form
z = tn u for some n ∈ ZZ and u ∈ O× .
(c) O is a principal ideal domain. More precisely, if P = tO and {0} = I ⊆ O
is an ideal then I = tn O for some n ∈ IN.
A ring having the above properties is called a discrete valuation ring. The
proof of Theorem 1.1.6 depends essentially on the following lemma.
Lemma 1.1.7. Let O be a valuation ring of the algebraic function field F/K,
let P be its maximal ideal and 0 = x ∈ P . Let x1 , . . . , xn ∈ P be such that
x1 = x and xi ∈ xi+1 P for i = 1, . . . , n − 1. Then we have
n ≤ [F : K(x)] < ∞ .
Proof. From Remark 1.1.2 and Proposition 1.1.5(c) follows that F/K(x) is
a finite extension, so it is sufficient to prove that x1 , . . . , xn are linearly
independent over K(x). Suppose there is a non-trivial linear combination
n
i=1 ϕi (x)xi = 0 with ϕi (x) ∈ K(x). We can assume that all ϕi (x) are
polynomials in x and that x does not divide all of them. Put ai := ϕi (0), the
constant term of ϕi (x), and define j ∈ {1, . . . , n} by the condition aj = 0 but
ai = 0 for all i > j. We obtain
−ϕj (x)xj =
ϕi (x)xi
(1.2)
i=j
with ϕi (x) ∈ O for i = 1, . . . , n (since x = x1 ∈ P ), xi ∈ xj P for i < j and
ϕi (x) = xgi (x) for i > j, where gi (x) is a polynomial in x. Dividing (1.2) by
xj yields
x
xi
ϕi (x) +
gi (x)xi .
−ϕj (x) =
x
x
j
i
i>j j
All summands on the right hand side belong to P , therefore ϕj (x) ∈ P . On
the other hand, ϕj (x) = aj + xgj (x) with gj (x) ∈ K[x] ⊆ O and x ∈ P , so
that aj = ϕj (x) − xgj (x) ∈ P ∩ K. Since aj = 0, this contradicts Proposition
1.1.5(c).
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1 Foundations of the Theory of Algebraic Function Fields
Proof of Theorem 1.1.6. (a) Assume that P is not principal, and choose an
element 0 = x1 ∈ P . As P = x1 O there is x2 ∈ P \ x1 O. Then x2 x−1
1 ∈ O,
x
∈
P
by
Proposition
1.1.5(b),
so
x
∈
x
P
.
By
induction
one
thereby x−1
1
1
2
2
obtains an infinite sequence x1 , x2 , x3 , . . . in P such that xi ∈ xi+1 P for all
i ≥ 1, a contradiction to Lemma 1.1.7.
(b) The uniqueness of the representation z = tn u with u ∈ O× is trivial, so
we only need to show the existence. As z or z −1 is in O we can assume that
z ∈ O. If z ∈ O× then z = t0 z. It remains to consider the case z ∈ P . There
is a maximal m ≥ 1 with z ∈ tm O, since the length of a sequence
x1 = z, x2 = tm−1 , x3 = tm−2 , . . . , xm = t
is bounded by Lemma 1.1.7. Write z = tm u with u ∈ O. Then u must be a unit
of O (otherwise u ∈ P = tO, so u = tw with w ∈ O and z = tm+1 w ∈ tm+1 O,
a contradiction to the maximality of m).
(c) Let {0} = I ⊆ O be an ideal. The set A := {r ∈ IN | tr ∈ I} is non-empty
(in fact, if 0 = x ∈ I then x = tr u with u ∈ O× and therefore tr = xu−1 ∈ I).
Put n := min (A). We claim that I = tn O. The inclusion I ⊇ tn O is trivial
since tn ∈ I. Conversely let 0 = y ∈ I. We have y = ts w with w ∈ O× and
s ≥ 0, so ts ∈ I and s ≥ n. It follows that y = tn · ts−n w ∈ tn O.
Definition 1.1.8. (a) A place P of the function field F/K is the maximal
ideal of some valuation ring O of F/K. Every element t ∈ P such that P =
tO is called a prime element for P (other notations are local parameter or
uniformizing variable).
(b) IPF := {P | P is a place of F/K}.
If O is a valuation ring of F/K and P is its maximal ideal, then O is
uniquely determined by P , namely O = {z ∈ F | z −1 ∈ P }, cf. Proposition
1.1.5(b). Hence OP := O is called the valuation ring of the place P .
A second useful description of places is given in terms of valuations.
Definition 1.1.9. A discrete valuation of F/K is a function v : F → ZZ∪{∞}
with the following properties :
(1) v(x) = ∞ ⇐⇒ x = 0 .
(2) v(xy) = v(x) + v(y) for all x, y ∈ F .
(3) v(x + y) ≥ min {v(x), v(y)} for all x, y ∈ F .
(4) There exists an element z ∈ F with v(z) = 1.
(5) v(a) = 0 for all 0 = a ∈ K.
In this context the symbol ∞ means some element not in ZZ such that
∞ + ∞ = ∞ + n = n + ∞ = ∞ and ∞ > m for all m, n ∈ ZZ. From (2) and
(4) it follows immediately that v : F → ZZ ∪ {∞} is surjective. Property (3) is
called the Triangle Inequality. The notions valuation and triangle inequality
are justified by the following remark:
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1.1 Places
5
Remark 1.1.10. Let v be a discrete valuation of F/K in the sense of Definition
1.1.9. Fix a real number 0 < c < 1 and define the function | |v : F → IR by
|z|v :=
cv(z)
0
if z = 0 ,
if z = 0 .
It is easily verified that this function has the properties of an ordinary absolute
value; the ordinary Triangle Inequality |x + y|v ≤ |x|v + |y|v turns out to be
an immediate consequence of condition (3) of Definition 1.1.9.
A stronger version of the Triangle Inequality can be derived from the
axioms and is often very useful:
Lemma 1.1.11 (Strict Triangle Inequality). Let v be a discrete valuation
of F/K and let x, y ∈ F with v(x) = v(y). Then v(x + y) = min {v(x), v(y)}.
Proof. Observe that v(ay) = v(y) for 0 = a ∈ K (by (2) and (5)), in particular
v(−y) = v(y). Since v(x) = v(y) we can assume v(x) < v(y). Suppose that
v(x + y) = min {v(x), v(y)}, so v(x + y) > v(x) by (3). Then we obtain
v(x) = v((x + y) − y) ≥ min {v(x + y), v(y)} > v(x), a contradiction.
Definition 1.1.12. To a place P ∈ IPF we associate a function vP : F →
ZZ∪{∞} (that will prove to be a discrete valuation of F/K) as follows: Choose
a prime element t for P . Then every 0 = z ∈ F has a unique representation
z = tn u with u ∈ OP× and n ∈ ZZ. Define vP (z) := n and vP (0) := ∞.
Observe that this definition depends only on P , not on the choice of t. In
fact, if t is another prime element for P then P = tO = t O, so t = t w for
some w ∈ OP× . Therefore tn u = (t n wn )u = t n (wn u) with wn u ∈ OP× .
Theorem 1.1.13. Let F/K be a function field.
(a) For a place P ∈ IPF , the function vP defined above is a discrete valuation
of F/K. Moreover we have
OP = {z ∈ F | vP (z) ≥ 0} ,
OP× = {z ∈ F | vP (z) = 0} ,
P = {z ∈ F | vP (z) > 0} .
(b) An element x ∈ F is a prime element for P if and only if vP (x) = 1.
(c) Conversely, suppose that v is a discrete valuation of F/K. Then the set
P := {z ∈ F | v(z) > 0} is a place of F/K, and OP = {z ∈ F | v(z) ≥ 0} is
the corresponding valuation ring.
(d) Every valuation ring O of F/K is a maximal proper subring of F .
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6
1 Foundations of the Theory of Algebraic Function Fields
Proof. (a) Obviously vP has the properties (1), (2), (4) and (5) of Definition
1.1.9. In order to prove the Triangle Inequality (3) consider x, y ∈ F with
vP (x) = n, vP (y) = m. We can assume that n ≤ m < ∞, thus x = tn u1
and y = tm u2 with u1 , u2 ∈ OP× . Then x + y = tn (u1 + tm−n u2 ) = tn z with
z ∈ OP . If z = 0 we have vP (x + y) = ∞ > min{n, m}, otherwise z = tk u
with k ≥ 0 and u ∈ OP× . Therefore
vP (x + y) = vP (tn+k u) = n + k ≥ n = min{vP (x), vP (y)} .
We have shown that vP is a discrete valuation of F/K. The remaining assertions of (a) are trivial, likewise (b) and (c).
(d) Let O be a valuation ring of F/K, P its maximal ideal, vP the discrete
valuation associated to P and z ∈ F \ O . We have to show that F = O[z].
To this end consider an arbitrary element y ∈ F ; then vP (yz −k ) ≥ 0 for
sufficiently large k ≥ 0 (note that vP (z −1 ) > 0 since z ∈ O ). Consequently
w := yz −k ∈ O and y = wz k ∈ O[z].
According to Theorem 1.1.13 places, valuation rings and discrete valuations
of a function field essentially amount to the same thing.
Let P be a place of F/K and let OP be its valuation ring. Since P is a
maximal ideal, the residue class ring OP /P is a field. For x ∈ OP we define
x(P ) ∈ OP /P to be the residue class of x modulo P , for x ∈ F \ OP we
put x(P ) := ∞ (note that the symbol ∞ is used here in a different sense
as in Definition 1.1.9). By Proposition 1.1.5 we know that K ⊆ OP and
K ∩ P = {0}, so the residue class map OP → OP /P induces a canonical
embedding of K into OP /P . Henceforth we shall always consider K as a
subfield of OP /P via this embedding. Observe that this argument also applies
˜ instead of K; so we can consider K
˜ as a subfield of OP /P as well.
to K
Definition 1.1.14. Let P ∈ IPF .
(a) FP := OP /P is the residue class field of P . The map x → x(P ) from F
to FP ∪ {∞} is called the residue class map with respect to P . Sometimes we
shall also use the notation x + P := x(P ) for x ∈ OP .
(b) deg P := [FP : K] is called the degree of P . A place of degree one is also
called a rational place of F/K.
The degree of a place is always finite; more precisely the following holds.
Proposition 1.1.15.
If P is a place of F/K and 0 = x ∈ P then
deg P ≤ [F : K(x)] < ∞ .
Proof. First we observe that [F : K(x)] < ∞ by Remark 1.1.2. Thus it suffices to show that any elements z1 , . . . , zn ∈ OP , whose residue classes
z1 (P ), . . . , zn (P ) ∈ FP are linearly independent over K, are linearly independent over K(x). Suppose there is a non-trivial linear combination
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1.1 Places
7
n
ϕi (x)zi = 0
(1.3)
i=1
with ϕi (x) ∈ K(x). W.l.o.g. we assume that the ϕi (x) are polynomials in x
and not all of them are divisible by x; i.e., ϕi (x) = ai + xgi (x) with ai ∈ K
and gi (x) ∈ K[x], not all ai = 0. Since x ∈ P and gi (x) ∈ OP , ϕi (x)(P ) =
ai (P ) = ai . Applying the residue class map to (1.3) we obtain
n
0 = 0(P ) =
n
ϕi (x)(P )zi (P ) =
i=1
ai zi (P ) .
i=1
This contradicts the linear independence of z1 (P ), . . . , zn (P ) over K.
˜ of constants of F/K is a finite field extension
Corollary 1.1.16. The field K
of K.
Proof. We use the fact that IPF = ∅ (which will be proved only in Corollary
˜ is embedded into FP via the residue
1.1.20). Choose some P ∈ IPF . Since K
˜
class map OP → FP , it follows that [K : K] ≤ [FP : K] < ∞.
Remark 1.1.17. Let P be a rational place of F/K; i.e., deg P = 1. Then we
have FP = K, and the residue class map maps F to K ∪ {∞} . In particular,
if K is an algebraically closed field, then all places are rational and we can
read an element z ∈ F as a function
z:
IPF
P
−→
−→
K ∪ {∞} ,
z(P ) .
(1.4)
This is why F/K is called a function field. The elements of K, interpreted
as functions in the sense of (1.4), are constant functions. For this reason K
is called the constant field of F . Also the following terminology is justified
by (1.4):
Definition 1.1.18. Let z ∈ F and P ∈ IPF . We say that P is a zero of z if
vP (z) > 0; P is a pole of z if vP (z) < 0. If vP (z) = m > 0, P is a zero of z
of order m; if vP (z) = −m < 0, P is a pole of z of order m.
Next we shall be concerned with the question as to whether there exist
places of F/K.
Theorem 1.1.19. Let F/K be a function field and let R be a subring of F
with K ⊆ R ⊆ F . Suppose that {0} = I
R is a proper ideal of R. Then
there is a place P ∈ IPF such that I ⊆ P and R ⊆ OP .
Proof. Consider the set
F := { S | S is a subring of F with R ⊆ S and IS = S } .
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1 Foundations of the Theory of Algebraic Function Fields
(IS is by definition the set of all finite sums
aν sν with aν ∈ I, sν ∈ S;
it is an ideal of S). F is non-empty as R ∈ F, and F is inductively ordered by inclusion. In fact, if H ⊆ F is a totally ordered subset of F then
T :=
{ S | S ∈ H } is a subring of F with R ⊆ T . We have to verify that
n
IT = T . Suppose this is false, then 1 =
ν=1 aν sν with aν ∈ I, sν ∈ T .
Since H is totally ordered there is an S0 ∈ H such that s1 , . . . , sn ∈ S0 , so
n
1 = ν=1 aν sν ∈ IS0 , a contradiction.
By Zorn’s lemma F contains a maximal element; i.e., there is a ring O ⊆ F
such that R ⊆ O ⊆ F , IO = O, and O is maximal with respect to these
properties. We want to show that O is a valuation ring of F/K.
As I = {0} and IO = O we have O F and I ⊆ O \ O× . Suppose there
exists an element z ∈ F with z ∈ O and z −1 ∈ O. Then IO[z] = O[z] and
IO[z −1 ] = O[z −1 ], and we can find a0 , . . . , an , b0 , . . . , bm ∈ IO with
1 = a0 + a1 z + · · · + an z n
and
(1.5)
1 = b0 + b1 z −1 + · · · + bm z −m .
(1.6)
Clearly n ≥ 1 and m ≥ 1. We can assume that m, n in (1.5) and (1.6) are
chosen minimally and m ≤ n. We multiply (1.5) by 1 − b0 and (1.6) by an z n
and obtain
1 − b0 = (1 − b0 )a0 + (1 − b0 )a1 z + · · · + (1 − b0 )an z n
0 = (b0 − 1)an z n + b1 an z n−1 + · · · + bm an z n−m .
and
Adding these equations yields 1 = c0 + c1 z + · · · + cn−1 z n−1 with coefficients
ci ∈ IO. This is a contradiction to the minimality of n in (1.5). Thus we have
proved that z ∈ O or z −1 ∈ O for all z ∈ F , hence O is a valuation ring of
F/K.
Corollary 1.1.20. Let F/K be a function field, z ∈ F transcendental over
K. Then z has at least one zero and one pole. In particular IPF = ∅.
Proof. Consider the ring R = K[z] and the ideal I = zK[z]. Theorem 1.1.19
ensures that there is a place P ∈ IPF with z ∈ P , hence P is a zero of z. The
same argument proves that z −1 has a zero Q ∈ IPF . Then Q is a pole of z.
Corollary 1.1.20 can be interpreted as follows: each z ∈ F , which is not in
˜ of F/K, yields a non-constant function in the sense of
the constant field K
Remark 1.1.17.
1.2 The Rational Function Field
For a thorough understanding of valuations and places in arbitrary function
fields, a precise idea of these notions in the simplest case is indispensable.
For this reason we investigate what these concepts mean in the case of the
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1.2 The Rational Function Field
9
rational function field F = K(x), where x is transcendental over K. Given an
irreducible monic polynomial p(x) ∈ K[x] we consider the valuation ring
Op(x) :=
f (x)
g(x)
f (x), g(x) ∈ K[x], p(x) g(x)
(1.7)
of K(x)/K with maximal ideal
Pp(x) =
f (x)
g(x)
f (x), g(x) ∈ K[x], p(x)|f (x), p(x) g(x)
.
(1.8)
In the particular case when p(x) is linear, i.e. p(x) = x − α with α ∈ K , we
abbreviate and write
Pα := Px−α ∈ IPK(x) .
(1.9)
There is another valuation ring of K(x)/K, namely
O∞ :=
f (x)
g(x)
f (x), g(x) ∈ K[x], deg f (x) ≤ deg g(x)
(1.10)
with maximal ideal
P∞ =
f (x)
g(x)
f (x), g(x) ∈ K[x], deg f (x) < deg g(x)
.
(1.11)
This place is called the infinite place of K(x). Observe that these labels
depend on the specific choice of the generating element x of K(x)/K (for
example K(x) = K(1/x), and the infinite place with respect to 1/x is the
place P0 with respect to x).
Proposition 1.2.1. Let F = K(x) be the rational function field.
(a) Let P = Pp(x) ∈ IPK(x) be the place defined by (1.8), where p(x) ∈ K[x]
is an irreducible polynomial. Then p(x) is a prime element for P , and the
corresponding valuation vP can be described as follows: if z ∈ K(x) \ {0} is
written in the form z = p(x)n · (f (x)/g(x)) with n ∈ ZZ, f (x), g(x) ∈ K[x],
p(x) f (x) and p(x) g(x), then vP (z) = n. The residue class field K(x)P =
OP /P is isomorphic to K[x]/(p(x)); an isomorphism is given by
φ:
K[x]/(p(x))
f (x) mod p(x)
−→
−→
K(x)P ,
f (x)(P ) .
Consequently deg P = deg p(x).
(b) In the special case p(x) = x − α with α ∈ K the degree of P = Pα is one,
and the residue class map is given by
z(P ) = z(α)
for z ∈ K(x) ,
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1 Foundations of the Theory of Algebraic Function Fields
where z(α) is defined as follows: write z = f (x)/g(x) with relatively prime
polynomials f (x), g(x) ∈ K[x]. Then
z(α) =
f (α)/g(α)
∞
if g(α) = 0 ,
if g(α) = 0 .
(c) Finally, let P = P∞ be the infinite place of K(x)/K defined by (1.11).
Then deg P∞ = 1. A prime element for P∞ is t = 1/x. The corresponding
discrete valuation v∞ is given by
v∞ (f (x)/g(x)) = deg g(x) − deg f (x) ,
where f (x), g(x) ∈ K[x]. The residue class map corresponding to P∞ is determined by z(P∞ ) = z(∞) for z ∈ K(x), where z(∞) is defined as usual:
if
z=
then
an xn + · · · + a0
with an , bm = 0 ,
bm xm + · · · + b0
⎧
⎪
⎨an /bm
z(∞) =
0
⎪
⎩
∞
if n = m ,
if n < m ,
if n > m .
(d) K is the full constant field of K(x)/K.
Proof. We prove only some essentials of this proposition; the remaining parts
of the proof are straightforward.
(a) Let P = Pp(x) , p(x) ∈ K[x] irreducible. The ideal Pp(x) ⊆ Op(x) is obviously generated by p(x), hence p(x) is a prime element for P . In order to prove
the assertion about the residue class field we consider the ring homomorphism
ϕ:
K[x]
f (x)
−→ K(x)P ,
−→ f (x)(P ) .
Clearly the kernel of ϕ is the ideal generated by p(x). Moreover ϕ is surjective:
if z ∈ Op(x) , we can write z = u(x)/v(x) with u(x), v(x) ∈ K[x] such that
p(x) v(x). Thus there are a(x), b(x) ∈ K[x] with a(x)p(x) + b(x)v(x) = 1,
therefore
a(x)u(x)
p(x) + b(x)u(x) ,
z =1·z =
v(x)
and z(P ) = (b(x)u(x))(P ) is in the image of ϕ. Thus ϕ induces an isomorphism
φ of K[x]/(p(x)) onto K(x)P .
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1.2 The Rational Function Field
11
(b) Now P = Pα with α ∈ K. If f (x) ∈ K[x] then (x − α)|(f (x) − f (α)),
hence f (x)(P ) = (f (x) − f (α))(P ) + f (α)(P ) = f (α). An arbitrary element
z ∈ OP can be written as z = f (x)/g(x) with polynomials f (x), g(x) ∈ K[x]
and (x − α) g(x) , therefore g(x)(P ) = g(α) = 0 and
z(P ) =
f (α)
f (x)(P )
=
= z(α) .
g(x)(P )
g(α)
(c) We will only show that 1/x is a prime element for P∞ . Clearly we have
that 1/x ∈ P . Consider some element z = f (x)/g(x) ∈ P∞ ; i.e., deg f < deg g.
Then
1 xf
, with deg(xf ) ≤ deg g .
z= ·
x g
This proves that z ∈ (1/x)O∞ , hence 1/x generates the ideal P∞ and it is
therefore a P∞ -prime element.
(d) Choose a place P of K(x)/K of degree one (e.g. P = Pα with α ∈ K). The
˜ of constants of K(x) is embedded into the residue class field K(x)P ,
field K
˜ ⊆ K(x)P = K.
hence K ⊆ K
Theorem 1.2.2. There are no places of the rational function field K(x)/K
other than the places Pp(x) and P∞ , defined by (1.8) and (1.11).
Corollary 1.2.3. The places of K(x)/K of degree one are in 1–1 – correspondence with K ∪ {∞} .
The corollary is obvious by Proposition 1.2.1 and Theorem 1.2.2. In terms
of algebraic geometry (cf. Appendix B) K ∪ {∞} is usually interpreted as
the projective line P1 (K) over K, hence the places of K(x)/K of degree one
correspond in a one-to-one way with the points of P1 (K).
Proof of Theorem 1.2.2. Let P be a place of K(x)/K. We distinguish two cases
as follows:
Case 1. Assume that x ∈ OP . Then K[x] ⊆ OP . Set I := K[x] ∩ P ; this
is an ideal of K[x] , in fact a prime ideal. The residue class map induces an
embedding K[x]/I → K(x)P , consequently I = {0} by Proposition 1.1.15.
It follows that there is a (uniquely determined) irreducible monic polynomial
p(x) ∈ K[x] such that I = K[x] ∩ P = p(x) · K[x]. Every g(x) ∈ K[x] with
p(x) g(x) is not in I , so g(x) ∈ P and 1/g(x) ∈ OP by Proposition 1.1.5. We
conclude that
Op(x) =
f (x)
f (x), g(x) ∈ K[x], p(x) g(x)
g(x)
⊆ OP .
As valuation rings are maximal proper subrings of K(x), cf. Theorem 1.1.13,
we see that OP = Op(x) .
Case 2. Now x ∈ OP . We conclude that K[x−1 ] ⊆ OP , x−1 ∈ P ∩ K[x−1 ]
and P ∩ K[x−1 ] = x−1 K[x−1 ]. As in case 1,
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1 Foundations of the Theory of Algebraic Function Fields
f (x−1 )
f (x−1 ), g(x−1 ) ∈ K[x−1 ], x−1 g(x−1 )
g(x−1 )
a0 + a1 x−1 + · · · + an x−n
b0 = 0
=
b0 + b1 x−1 + · · · + bm x−m
a0 xm+n + · · · + an xm
b0 = 0
=
b0 xm+n + · · · + bm xn
u(x)
u(x), v(x) ∈ K[x], deg u(x) ≤ deg v(x)
=
v(x)
= O∞ .
OP ⊇
Thus OP = O∞ and P = P∞ .
1.3 Independence of Valuations
The main result of this section is the Weak Approximation Theorem 1.3.1
(which is also referred to as the Theorem of Independence). Essentially this
says the following: If v1 , . . . , vn are pairwise distinct discrete valuations of
F/K and z ∈ F , and if we know the values v1 (z), . . . , vn−1 (z), then we cannot
conclude anything about vn (z). A substantial improvement of Theorem 1.3.1
will be given later in Section 1.6.
Theorem 1.3.1 (Weak Approximation Theorem). Let F/K be a function field, P1 , . . . , Pn ∈ IPF pairwise distinct places of F/K, x1 , . . . , xn ∈ F
and r1 , . . . , rn ∈ ZZ. Then there is some x ∈ F such that
vPi (x − xi ) = ri for i = 1, . . . , n .
Corollary 1.3.2. Every function field has infinitely many places.
Proof of Corollary 1.3.2. Suppose there are only finitely many places, say
P1 , . . . , Pn . By Theorem 1.3.1 we find a non-zero element x ∈ F with vPi (x) >
0 for i = 1, . . . , n. Then x is transcendental over K since it has zeros. But x
has no pole; this is a contradiction to Corollary 1.1.20.
Proof of Theorem 1.3.1. The proof is somewhat technical and therefore divided
into several steps. For simplicity we write vi instead of vPi .
Step 1. There is some u ∈ F with v1 (u) > 0 and vi (u) < 0 for i = 2, . . . , n.
Proof of Step 1. By induction. For n = 2 we observe that OP1 ⊆ OP2 and vice
versa, since valuation rings are maximal proper subrings of F , cf. Theorem
1.1.13. Therefore we can find y1 ∈ OP1 \ OP2 and y2 ∈ OP2 \ OP1 . Then
v1 (y1 ) ≥ 0, v2 (y1 ) < 0, v1 (y2 ) < 0 and v2 (y2 ) ≥ 0. The element u := y1 /y2
has the property v1 (u) > 0, v2 (u) < 0 as desired.
For n > 2 we have by induction hypothesis an element y with v1 (y) > 0,
v2 (y) < 0, . . . , vn−1 (y) < 0. If vn (y) < 0 the proof is finished. In case vn (y) ≥ 0
www.pdfgrip.com
1.3 Independence of Valuations
13
we choose z with v1 (z) > 0, vn (z) < 0 and put u := y + z r . Here r ≥ 1 is
chosen in such a manner that r · vi (z) = vi (y) for i = 1, . . . , n − 1 (this
is obviously possible). It follows that v1 (u) ≥ min{v1 (y), r · v1 (z)} > 0 and
vi (u) = min{vi (y), r · vi (z)} < 0 for i = 2, . . . , n (observe that the Strict
Triangle Inequality applies).
Step 2. There is some w ∈ F such that v1 (w − 1) > r1 and vi (w) > ri for
i = 2, . . . , n.
Proof of Step 2. Choose u as in Step 1 and put w := (1 + us )−1 . We have, for
sufficiently large s ∈ IN, v1 (w − 1) = v1 (−us (1 + us )−1 ) = s · v1 (u) > r1 , and
vi (w) = −vi (1 + us ) = −s · vi (u) > ri for i = 2, . . . , n.
Step 3. Given y1 , . . . , yn ∈ F , there is an element z ∈ F with vi (z −yi ) > ri
for i = 1, . . . , n.
Proof of Step 3. Choose s ∈ ZZ such that vi (yj ) ≥ s for all i, j ∈ {1, . . . , n}.
By Step 2 there are w1 , . . . , wn with
vi (wi − 1) > ri − s and
vi (wj ) > ri − s for j = i .
n
Then z := j=1 yj wj has the desired properties.
Now we are in a position to finish the proof of Theorem 1.3.1. By Step
3 we can find z ∈ F with vi (z − xi ) > ri , i = 1, . . . , n. Next we choose zi
with vi (zi ) = ri (this is trivially done). Again by Step 3 there is z with
vi (z − zi ) > ri for i = 1, . . . , n. It follows that
vi (z ) = vi ((z − zi ) + zi ) = min{vi (z − zi ), vi (zi )} = ri .
Let x := z + z . Then
vi (x − xi ) = vi ((z − xi ) + z ) = min{vi (z − xi ), vi (z )} = ri .
In Section 1.4 we shall show that an element x ∈ F which is transcendental
over K has as many zeros as poles (counted properly). An important step
towards that result is our next proposition which sharpens both of Lemma
1.1.7 and Proposition 1.1.15. The Weak Approximation Theorem will play a
significant role in the proof.
Proposition 1.3.3. Let F/K be a function field and let P1 , . . . , Pr be zeros
of the element x ∈ F . Then
r
vPi (x) · deg Pi ≤ [F : K(x)] .
i=1
Proof. We set vi := vPi , fi := deg Pi and ei := vi (x). For all i there is an
element ti with
vi (ti ) = 1 and vk (ti ) = 0 for k = i .
