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 IB G O R D O N A N D B R E A C H - . SCIENCE PUBLISHERS NEW YORK LONDON PARIS Copyright 0 I967 by Gordon and Breach, Science Publishers, Inc. General Preface 150 Fifth Avenue, New York, New York 10011 Library of Congress Catalog Card Number: 67-26811 A large number of mathematical books begin as lecture notes; Editorial Ofice for Great Britain: but, since mathematicians are busy, and since the labor required to bring lecture notes up to the level of perfection which authors Gordon and Breach Science Publishers Ltd. and the public demand of formally published books is very 8 Bloomsbury Way considerable, it follows that an even larger number of lecture London WCI, England notes make the transition to book form only after great delay or not at all. The present lecture note series aims to fill the resulting Editorial Ofice for France: gap. It will consist of reprinted lecture notes, edited at least to a 7-9 rue Emile Dubois satisfactory level of completeness and intelligibility, though not Paris 14e necessarily to the perfection which is expected of a book. In addition to lecture notes, the series will include volumes of collected Distributed in France by: reprints of journal articles as current developments indicate, and Dunod Editeur mixed volumes including both notes and reprints. 92, rue Bonaparte Paris 6e Distributed in Canada by: The Ryerson Press 299 Queen Street West Toronto 2B, Ontario Printed in Belgium by the Saint Catherine Press, Ltd., Tempelhof, Bruges 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 valuations. In terms of this function we may define a topology in k by

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