ALGEBRAIC NUMBERS AND ALGEBRAIC FUNCTIONS

183 Pages·2004·8.38 MB·English

ALGEBRAIC NUMBERS AND
ALGEBRAIC FUNCTIONS
Notes on Mathematics and Its Applications
General Editors: Jacob T. Schwartx, Courant Institute of Mathe- Algebraic Numbers and
matical Sciences and Maurice Lbvy, Universitk de Paris
E. Artin, ALGEBRAIC NUMBERS AND ALGEBRAIC FUNCTIONS Algebraic Functions
R. P. Boas, COLLECTED WORKS OF HIDEHIKO YAMABE
M. Davis, FUNCTIONAL ANALYSIS
M. Davis, LECTURES ON MODERN MATHEMATICS
J. Eells, Jr., SINGULARITIES OF SMOOTH MAPS
K. 0. Friedrichs, ADVANCED ORDINARY DIFFERENTIAL EQUATIONS
K. 0. Friedrichs, SPECIAL TOPICS IN FLUID DYNAMICS
K. 0. Friedrichs and H. N. Shapiro, INTEGRATION IN HILBERT SPACE
M. Hausner and J. T. Schwartz, LIE GROUPS; LIE ALGEBRAS
P. Hilton, HOMOTOPY THEORY AND DUALITY
I;: John, LECTURES ON ADVANCED NUMERICAL ANALYSIS
Allen M. Krall, STABILITY TECHNIQUES
H. Mullish, AN INTRODUCTION TO COMPUTER PROGRAMMING
J. T. Schwartx, w* ALGEBRAS
A. Silverman, EXERCISES IN FORTRAN
J. J. Stoker, NONLINEAR ELASTICITY
EMIL ARTIN
Late of Princeton University
Additional volumes in preparation
GI
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Preface
These lecture notes represent the content of a course given at
Princeton University during the academic year 1950151. This
course was a revised and extended version of a series of lectures
given at New York University during the preceding summer.
They cover the theory of valuation, local class field theory, the
elements of algebraic number theory and the theory of algebraic
function fields of one variable. It is intended to prepare notes for
a second part in which global class field theory and other topics
will be discussed.
Apart from a knowledge of Galois theory, they presuppose a
sufficient familiarity with the elementary notions of point set
topology. The reader may get these notions for instance in
N. Bourbaki, Eltments de Mathtmatique, Livre III, Topologie
gtntrale, Chapitres 1-111.
In several places use is made of the theorems on Sylow groups.
For the convenience of the reader an appendix has been prepared,
containing the proofs of these theorems.
The completion of these notes would not have been possible
without the great care, patience and perseverance of Mr. I. T. A. 0.
Adamson who prepared them. Of equally great importance have
been frequent discussions with Mr. J. T. Tate to whom many
simplifications of proofs are due. Very helpful was the assistance
of Mr. Peter Ceike who gave a lot of his time preparing the stencils
for these notes.
Finally I have to thank the Institute for Mathematics and
Mechanics, New York University, for mimeographing these notes.
Princeton University
June 1951
Contents
v
vii
Part I General Valuation Theory
Chapter 1
VALUATIONOSF A FIELD
1. Equivalent Valuations . . . . . . . .
2. The Topology Induced by a Valuation . . . .
3. Classification of Valuations . . . . . . .
4. The Approximation Theorem . . . . . .
5. Examples . . . . . . . . .
6. Completion of a Field . . . . . . . .
Chapter 2
COMPLETFEI ELDS
1. Normed Linear Spaces . . . . . . .
2. Extension of the Valuation . . . . .
3. Archimedean Case . . . . . . . . .
4. The Non-Archimedean Case . . . . . .
5. Newton's Polygon . . . . . . . .
6. The Algebraic Closure of a Complete Field . . .
7. Convergent Power Series . . . . . . .
Chapter 3
1. The Ramification and Residue Class Degree . . .
2. The Discrete Case . . . . . . . .
3. The General Case . . . . . . . .
CONTENTS CONTENTS
Chapter 4 Chapter 9
RAMIFICATIOTNH EORY THEE XISTENCTEH EOREM
I. Unramified Extensions . . . . 1. Introduction . . . . . . . . . .
2. Tamely Ramified Extensions . 2. The Infinite Product Space I . . . . .
3. Characters of Abelian Groups . . . 3. The New Topology in K* . . . . . .
4. The Inertia Group and Ramification Group 4. The Norm Group and Norm Residue Symbol for
5. Higher Ramification Groups . Infinite Extensions . . . . . . . .
6. Ramification Theory in the Discrete Case . 5. Extension Fields with Degree Equal to the Characteristic
6. The Existence Theorem . . . . . . .
Chapter 5 7. Uniqueness of the Norm Residue Symbol . .
THED IFFERENT
1. The Inverse Different . . . . Chapter 10
2. Complementary Bases . . . . APPLICATIONASN D ILLUSTRATIONS
3. Fields with Separable Residue Class Field 1. Fields with Perfect Residue Class Field . . . .
4. The Ramification Groups of a Subfield .
2. The Norm Residue Symbol for Certain Power Series
Fields . . . . . . . . . .
Part ll Local Class Field Theory 3. Differentials in an Arbitrary Power Series Field . .
.
4. The Conductor and Different for Cyclic p-Extensions
Chapter 6
5. The Rational p-adic Field . . . . . . .
PREPARATIOFNOSR LOCALC LASSF IELD THEORY 6. Computation of the Index (a : an) . . . . .
1. Galois Theory for Infinite Extensions . .
2. Group Extensions . . . .
3. Galois Cohomology Theory . . . . Part II I
4. Continuous Cocycles . . . . Product Formula and Function Fields in One Variable
Chapter 7 Chapter 11
THEF IRSTA ND SECONDIN EQUALITIES
1. Introduction . . . . . 1. The Radical of a Ring . . . . . . . 2 1 5
2. Unramified Extensions . . . 2. Kronecker Products of Spaces and Rings . . . . 216
3. The First Inequality . . . . 3. Composite Extensions . . . . . . . 2 1 8
4. The Second Inequality: A Reduction Step . 4. Extension of the Valuation of a Non-Complete Field . 223
5. The Second Inequality Concluded . . .
Chapter 12
Chapter 8 CHARACTERIZATOIFO NFI ELDSB Y THE PRODUCFTO RMULA
THEN ORMR ESIDUES YMBOL 1. PF-Fields. . . . . . . . . 2 2 5
1. The Temporary Symbol (c, I KIT) . . . 2. Upper Bound for the Order of a Parallelotope . . . 227
K
2. Choice of a Standard Generator c . . . 3. Description of all PF-Fields . . . . . . 230
3. The Norm Residue Symbol for Finite Extensions 4. Finite Extensions of PF-Fields . . . . . . 235
xii CONTENTS CONTENTS
Chapter 13 Chapter 17
DIFFERENTIAILNS PF-FIELDS DIFFERENTIAILNS FUNCTIONF IELDS
. . . . . . . .
1. Valuation Vectors. Ideles. and Divisors . . . 2 3 8 I Prepa.rations. . 321
2 . Valuation Vectors in an Extension Field . 241 2. Local Components of Differentials . . . . . 32 2
3 . Some Results on Vector Spaces . . . . 2 4 4 3 Differentials and .Derivatives in Func.tion Fields . 324
4 . Differentials in the Rational Subfield of a PF-Field . . 245 4. Differentials of the First Kind . . . . . 3 2 9
5. Differentials in a PF-Field . . . . . 2 5 1
6 . The Different . . . . . 255 Appendix
THEOREOMN Sp -GROUPSA ND SYLOWG ROUPS
Chapter 14 1. S-Equivalence Classes . . . . 3 3 4
THER IEMANN-ROCTHH EOREM 2 . Theorems About $-Groups . . . . . . . 33 5
3. The Existence of Sylow Subgroups . . . . . 33 6
1. Parallelotopes in a Function Field . . 260 4 . Theorems About Sylow Subgroups . . . . . 33 7
2 . First Proof . . . . . . 2 6 2
3. Second Proof . . . . . . . 2 6 5
Chapter 15
CONSTANFTI ELDE XTENSIONS
1. The Effective Degree . . . . . . 2 7 1
2. Divisors in an Extension Field . . . . . 2 7 8
3 . Finite Algebraic Constant Field Extensions . . . 279
.
4 The Genus in a Purely Transcendental Constant Field
Extension . . . . . . . . . 2 8 4
5 . The Genus in an Arbitrary Constant Field Extension . 287
Chapter 16
APPLICATIONOFS THE RIEMANN-ROCTHH EOREM
1. Places and Valuation Rings . . . . 293
2 . Algebraic Curves . . . . 2 9 7
3. Linear Series . . . . . . 3 0 0
4 . Fields of Genus Zero . . . 302
5. Elliptic Fields . . . . . . . . . 30 6
,
6 . The Curve of Degree n . . 3 1 1
7. Hyperelliptic Fields . . . . . . 3 1 2
8. The Theorem of Clifford . . . . 317
PART ONE
General Valuation Theory
CHAPTER ONE
Valuations of a Field
A vahation of a field k is a real-valued function I x I , defined
for all x E k, satisfying the following requirements:
(I) I x I 2 0; ( x ( = 0 if and only if x = 0,
(2) IXYI = 1x1 ly I,
< + <
(3) If I x I 1, then I 1 x I c, where c is a constant;
c 2 1.
(1) and (2) together imply that a valuation is a homomorphism of
the multiplicative group k* of non-zero elements of k into the
positive real numbers.
If this homomorphism is trivial, i.e. if I x [ = 1 for all x E k*,
the valuation is also called trivial.
1. Equivalent Valuations
Let I I, and I ,1 be two functions satisfying conditions (I) and
(2) above; suppose that ( ,1 is non-trivial. These functions are
said to be equivalent if I a ,1 < 1 implies I a 1, < 1. Obviously for
such functions ( a ,1 > 1 implies 1 a ,1 > 1; but we can prove
more.
Theorem 1: Let I ,1 and I ,1 be equivalent functions, and
suppose I ,1 is non-trivial. Then 1 a 1, = 1 implies ( a ,1 = 1.
Proof: Let b # 0 be such that I b 1, < 1. Then I anb 1, < 1;
whence 1 anb ,1 < I, and so 1 a 1, < 1 b ];lln. Letting n +a,
<
we have I a 1, 1. Similarly, replacing a in this argument by lla,
we have I a 1, 2 1, which proves the theorem.
3
4 1. VALUATIONS OF A FIELD 2. THE TOPOLOGY INDUCED BY A VALUATION
Corollary: For non-trivial functions of this type, the relation
of equivalence is reflexive, symmetric and transitive.
There is a simple relation between equivalent functions, given
Now given a,, a,, a,, we can find an integer r such that
by < a*.,
n 2' < 2n. Hence
Theorem 2: If j 1, and / 1, are equivalent functions, and I II
is non-trivial, then I a 1, = I a lla for all a E k, where or is a fixed
positive real number.
Proof: Since 1 1, is non-trivial, we can select an element In particular, if we set all the a, = 1, we have I n 1 < 2n. We may
c E k* such that I c 1, > 1; then I c 1, > 1 also. also weaken the above inequality, and write
Set 1 a 1, = 1 c l,Y, where y is a non-negative real number. If
m/n > y, then 1 a 1, < 1 c llmln, whence 1 an/cm1 , < 1. Then
I an/cm1 , < 1, from which we deduce that I a 1, < I c I,mln. Simi-
larly, if m,n < y, then I a 1, > 1 c ImJn. It follows that I a ,1 = I c Jk.
Now, clearly,
1% l a I, log I a I,
Y=logJcJ,=log~
This proves the theorem, with
In view of this result, let us agree that the equivalence class
defined by the trivial function shall consist of this function alone.
Our third condition for valuations has replaced the classical
+ < + .
"Triangular Inequality" condition, viz., 1 a b 1 1 a 1 I b 1
The connection between this condition and ours is given by Letting n +co we obtain the desired result.
We may note that, conversely, the triangular inequality implies
Theorem 3: Every valuation is equivalent to a valuation for that our third requirement is satisfied, and that we may choose
which the triangular inequality holds. c = 2.
Proof. (1) When the constant c = 2, we shall show that the (2) When c > 2, we may write c = 2". Then it is easily verified
triangular inequality holds for the valuation itself. that ( Illa is an equivalent valuation for which the triangular
Let ] a ]G j b j . inequality is satisfied.
Then
2. The Topology Induced by a Valuation
Similarly Let I I be a function satisfying the axioms (1) and (2)f or valua-
tions. In terms of this function we may define a topology in k by