AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 9

The Theory of Algebraic Second Edition

Harry Pollard and Harold G. Diamond 10.1090/car/009

THE THEORY OF ALGEBRAIC NUMBERS

SECOND EDITION

By

HARRY POLLARD and HAROLD G. DIAMOND THE CARUS MATHEMATICAL MONOGRAPHS

Published by THE MATHEMATICAL ASSOCIATION OF AMERICA

Editorial Committee

PHILIP FRANKLIN, Chairman E. F. BECKENBACH Η. S. Μ. COXETER Ν. Η. McCOY KARL MENGER J. L. SYNGE HE CARUS METHEMATICAL MONOGRAPHS are an T expression of the desire of Mrs. Mary Hegeler Cams, and of her son, Dr. Edward H. Cams, to contribute to the dissemination of mathematical knowledge by making accessible at nominal cost a series of expository presentations of the best thoughts and keenest researches in pure and applied mathemat- ics. The publication of the first four of these monographs was made possible by a notable gift to the Mathematical Association of America by Mrs. Cams as sole trustee of the Edward C. Hegeler Trust Fund. The sales from these have resulted in the Carus Monograph Fund, and the Mathematical Association has used this as a revolving book fund to publish the succeeding monographs. The expositions of mathematical subjects which the mono- graphs contain are forth in a manner comprehensible not only to teachers and students specializing in mathematics, but also to scientific workers in other fields, and especially to the wide of thoughtful people who, having a moderate acquaintance with elementary mathematics, wish to extend their knowledge with- out prolonged and critical study of the mathematical journals and treatises. The scope of this series includes also historical and biographical monographs.

The following monographs have been published:

No. 1. Calculus of Variations, G. A. BLISS

No. 2. Analytic Functions of a Complex Variable, by D. R. CURTISS

No. 3. Mathematical Statistics, by H. L. RIETZ

No. 4. Projective Geometry, by J. W. YOUNG

No. 5. A History of Mathematics in America before 1900, by D. E. SMITH and JEKUTHIEL GINSBURG (out of print)

No. 6. Fourier Series and Orthogonal , by DUNHAM JACKSON

No. 7. Vectors and Matrices, by C. C. MACDUFFEE

No. 8. Rings and Ideals, by Ν. Η. McCOY No. 9. The Theory of Algebraic Numbers, Second edition, by HARRY POLLARD and HAROLD G. DIAMOND

No. 10. The Theory of Quadratic Forms, by B.W.JONES

No. 11. Irrational Numbers, by IVAN NIVEN

No. 12. Statistical Independence in Probability, Analysis and Theory, by MARK KAC

No. 13. A Primer of Real Functions, Third edition, by RALPH P. BOAS, JR.

No. 14. Combinatorial Mathematics, by HERBERT JOHN RYSER

No. 15. Noncommutative Rings, by I. N. HERSTEIN

No. 16. Dedekind Sums, by HANS RADEMACHER and EMIL GROSSWALD

No. 17. The Schwarz Function and its Applications, by PHILIP J. DAVIS

No. 18. Celestial Mechanics, by HARRY POLLARD

No. 19. Theory and its Classical Problems, by CHARLES ROBERT HADLOCK

No. 20. The Generalized Riemann Integral, by ROBERT M. McLEOD

No. 21. From Error-Correcting Codes through Sphere Packings to Simple Groups, by THOMAS M. THOMPSON.

No. 22. Random Walks and Electric Networks, by PETER G. DOYLE and J. LAURIE SNELL The Cams Mathematical Monographs

NUMBER NINE

THE THEORY OF ALGEBRAIC NUMBERS

SECOND EDITION

By HARRY POLLARD

Purdue University

and

HAROLD G. DIAMOND University of Illinois

Published and distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA Second Edition © 1975 by The Mathematical Association of America (Incorporated)

Library of Congress Catalog Card Number 75-27003

Paperback ISBN 978-0-88385-138-8 eISBN 978-1-61444-009-3 Hardcover (second edition, out of print) ISBN 978-0-88385-018-3 Hardcover (first edition, out of print) ISBN 9788-0-88385-009-1

Printed in the United States of America

Current printing (last digit): 10 9 8 7 6 5 4 3 DEDICATED TO PROFESSOR DAVID VERNON WIDDER CONTENTS

CHAPTER I. Divisibility 1. Uniqueness of 1 2. A general problem 5 3. The Gaussian 7 Problems H II. The Gaussian Primes 1. Rational and Gaussian primes 14 2. Congruences 14 3. Determination of the Gaussian primes 18 4. Fermat's theorem for Gaussian primes 21 Problems 23 III. Polynomials over a field 1. The of polynomials 25 2. The Eisenstein irreducibility criterion 30 3. Symmetric polynomials 35 Problems 40 IV. Algebraic Number Fields 1. Numbers algebraic over a field 44 2. Extensions of a field 46 3. Algebraic and transcendental numbers 51 Problems 56 V. Bases 1. Bases and finite extensions 59 2. Properties of finite extensions 62 3. Conjugates and discriminants 64 4. The 68 Problems 70 VI. Algebraic Integers and Integral Bases 1. Algebraic integers 74 2. The integers in a quadratic field 77 3. Integral bases 79 4. Examples of integral bases 82 Problems 86 CONTENTS

CHAPTER VII. Arithmetic in Algebraic Number Fields 1. Units and primes 88 2. Units in a quadratic field 90 3. The uniqueness of factorisation 93 4. Ideals in an 95 Problems 98 VIII. The Fundamental Theorem of Ideal Theory 1. Basic properties of ideals 102 2. The classical proof of the unique factorization theorem 107 3. The modern proof 112 Problems 116 IX. Consequences of the Fundamental Theorem 1. The highest common factor of two ideals 120 2. Unique factorization of integers 123 3. The problem of ramification 125 4. Congruences and norms 128 5. Further properties of norms 132 Problems 135 X. Ideal Classes and Class Numbers 1. Ideal classes 139 2. Class numbers 140 Problems 1*4 XI. The Fermat Conjecture 1. Pythagorean triples 146 2. The Fermat conjecture 147 3. Units in cyclotomic fields 150 4. Kummer's theorem 155 Problems 158

References 159 List of Symbols 160 Index 161

Preface to First Edition The purpose of this monograph is to make available in English the elementary parts of classical algebraic . An earlier version in mimeographed form was used at Cornell University in the spring term of 1947-48, and the present version has accordingly profited from the criticisms of several readers. I am particularly indebted to Miss Leila R. Raines for her painstaking assistance in the revision and preparation of the manuscript for publication.

Preface to Second Edition This new edition follows closely the plan and style of the 1st edition. The principal changes are the correction of misprints, the expansion or simplification of some argu- ments, and the omission of the final chapter on units in order to make way for the introduction of some two hundred problems. The credit for this revision is entirely due to my associate, Harold G. Diamond. We are grateful to many contributors for both correc- tions and problems: P. T. Bateman, B. C. Berndt, P. T. Montague, Ivan Niven, J. Steinig, and S. V. Ullom.

HARRY POLLARD August 1974

LIST OF SYMBOLS with page on which defined

1 3, 6, 26, 103. Ä(i) 40.

0 6. 46.

J 6. F(«i, ... , a,) 47. 6 6. (K/F) 60.

Η 6. |α,·,·| 61. a 7. Δ[αι α„] 67. Ν 7, 89, 129. D(ß) 68. => 12. (αϊ, ..., at) 97. = 15, 128. (α) 97. mod 15, 128. D 103.

(Λ, fc) 16. Ρ"1 113. ~ 19, 139. 04,fi) 120. R 25. C° 120η. C 25. S(e) 126.

F[x] 26. A + Β 135. Jlx] 26. Α Π ß 136. /*(x) 32. [α] 150.

σ, 36. L 150.

160 INDEX

Algebraic extension of field, 46, Extension of field, 46; simple 47. algebraic, 46; multiple alge- Algebraic , 74; number, braic, 47; finite, 60, 62. 44, 51. Factor, 1,103. Associate, 19, 88. Factorization, uniqueness of, 93. Baker, Α., 56. Fermat Theorem, 16, 21, 134; Basis, 59; of ideal, 95; integral, conjecture (= "Last 79; minimal, 82. Theorem"), 146. Borevich, Ζ. I., 149. Field, algebraic number, 44, 64; Class number, 139. cyclotomic, 68; Euclidean, Cummutative ring, 25. 125; finite, 42; imaginary, Complete residue system, 16, 91; non-real, 91; number, 133. 25; quadratic, 77. Congruence of Gaussian integers, Field , 65. 21; modulo an ideal, 128; Fractional ideal, 114. of numbers, 15. Fundamental theorem of alge- Conjugate of algebraic number, bra, 27; of arithmetic, 1, 5; 45; complex, 7; for a field, 65. of ideal theory, 102, 112, Cyclotomic polynomial, 34. 116. Dedekind-Hasse criterion, 124. Gauss, C. F., 125; Lemma, 31. Degree of algebraic number, 45; , 6; prime, 14; of extension, 60; of poly- prime classification, 18. nomial, 26. Gelfond-Schneider Theorem, 55. Discriminant of basis, 67; of Generator of an ideal, 97. field, 82. Greatest common divisor of Divide, 3, 6, 26, 88, 103. numbers, 16; of poly- Divisor of ideal, 103. nomials, 41. Eisenstein's irreducibility cri- Group, 25. terion, 30, 33. Hardy, G. H., 11, 55,125. Elementary symmetric function, Hecke, Ε., 93. 36. Heilbronn, Η., 125. Equivalence relation, 57. Highest common factor of two Equivalent ideals, 139. ideals, 120. Euclid, 14. Hille, Ε., 55. Euclidean algorithm, 12. Hurwitz, Α., 85. 162 INDEX

Ideal, 95; in F[x], 56; irreducible, Prime, regular, 149; relatively 105; maximal, 105; prime, (see relatively prime). 105, 116; principal, 97; Primitive number triple, 146; ramified, 126. polynomial, 30; root of Ideal class, 139. unity, 69. Ideals, equivalent, 139. Principal ideal, 97. Infinitude of primes, 14, 90. Product of ideals, 103. Integer, algebraic, 74; Gaussian, Pythagorean triple, 146. 6; in quadratic field, 79; Quadratic field, 77. rational, 6. Ramified ideal, 126. Integral basis, 79. Rational integer, 6; root Irreducible ideal, 105; poly- theorem, 41. nomial, 27. Regular prime, 149. Krull, W., 85. Relatively prime ideals, 120; Kummer, Ε., 149; Theorem, 155. rational integers, 2; Gaus- Landau, E., 55, 77, 149. sian integers, 21; poly- Lang, S., 56. nomials, 28. Least common multiple of two Residue class, 128; system, 16. ideals, 136. Ring, 25. LeVeque, W. J., 53. Root of polynomial, 27; of unity, Lexicographic order, 37. 69. Linear independence, 59. Roth, K. F., 53. Linfoot, E., 125. Shafarevich, I. R., 149. Liouville, J., 53. Stark, Η. M., 125. Maximal ideal, 105. Symmetric polynomial, 35; ho- Minimal polynomial, 44. mogeneous, 36. Minkowski, H., 140. Thomas, J. M., 50. , 30. Trace, 126. Noether, E., 85. , 51. Norm of ideal, 129; of algebraic Unique factorization of ideals, integer, 89; of Gaussian 105, 112; of integers, 1, 5, integer, 7. 10, 123. Number field, 25. Unit, 5, 6, 28, 88; in quadratic Ore, O., 112. field, 90. Polynomial, 26; cyclotomic, 34; Unramified ideal, 126. field, 65; irreducible, 27; Uspensky, J. V., 68. minimal, 44; monic, 30; Vandermonde determinant, 68. prime, 27; primitive, 30; Vandiver, H. S., 149. symmetric, 35. Wilson's Theorem, 17. Prime ideal, 105, 116; number, Wright, Ε. M., 11, 55, 125. 1, 6, 14, 88. Zero of a polynomial, 27. AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS

This monograph makes available in English the elementary parts of classical .

This second edition follows closely the plan and style of the 1st edition. The principal changes are the correction of misprints, the expansion or simplifi cation of some arguments, and the omission of the fi nal chapter on units in order to make way for the introduction of some two hundred problems.