FUNDAMENTALS OF MATHEMATICS VOLUME I Foundations of Mathematics The Real Number System and Algebra Fundamentals of Mathematics Volume I Foundations of Mathematics The Real Number System and Algebra n Volume Geometry Volume III Analysis FUNDAMENTALS OF MATHEMATICS VOLUME I Foundations of Mathematics The Real Number System and Algebra Edited by H. Behnke F. Bachmann K. Fladt w. Suss with the assistance 9f H. Gerike F. Hohenberg G. Pickert H. Rau Translated by S. H. Gould The MIT Press Cambridge, Massachusetts, and London, England Originally published by Vandenhoeck & Ruprecht, Gottingen, Germany, under the title Grundzuge der Mathematik. The publication was sponsored by the German section of the International Commission for Mathematical Instruction. The translation of this volume is based upon the second German edition of 1962. Third printing, 1986 First MIT Press paperback edition, 1983 English translation copyright © 1974 by The Massachusetts Institute of Technology. Printed and bound in the United States of America. All rights reserved. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording, or by any informa tion storage and retrieval system, without permission in writing from the publisher. ISBN 0-262-02048-3 (hardcover) 0-262-52093-1 (paperback) Library of Congress catalog card number: 68-14446 Contents Translator's Foreword ix From the Preface (to the 1958 Edition), Heinrich Behnke and Kuno Fladt x PART A FOUNDATIONS OF MATHEMATICS H. Hermes and W. Markwald 1. Conceptions of the Nature of Mathematics 3 2. Logical Analysis of Propositions 9 3. The Concept of a Consequence 20 4. Axiomatization 26 5. The Conc~pt of an Algorithm 32 6. Proofs 41 7. Theory of Sets 50 8. Theory of Relations 61 9. Boolean Algebra 66 10. Axiomatization of the Natural Numbers 71 1 I. Antinomies 80 Bibliography 86 PART B 89 ARITHMETIC AND ALGEBRA Introduction, W. Grobner 91 CHAPTER I Construction of the System of Real Numbers, G. Pickert and L. Gorke 93 1. The Natural Numbers 93 v vi CONTENTS 2. The Integers 105 3. The Rational Numbers 121 4. The Real Numbers 129 Appendix: Ordinal Numbers, D. Kurepa and A. Aymanns 153 CHAPTER 2 Groups, W. Gaschutz and H. Noack 166 1. Axioms and Examples ]67 2. Immediate Consequences of the Axioms for a Group 178 3. Methods of Investigating the Structure of Groups 182 4. Isomorphisms 188 5. Cyclic Groups 19) 6. Normal Subgroups and Factor Groups 194 7. The Commutator Group 197 8. Direct Products 198 9. Abelian Groups ]99 10. The Homomorphism Theorem 212 11. The Isomorphism Theorem 214 12. Composition Series, Jordan-HOlder Theorem 215 13. Normalizer, Centralizer, Center 217 14. p-Groups 219 15. Permutation Groups 220 16. Some Remarks on More General Infinite Groups 230 CHAPTER 3 Linear Algebra, H. Gericke and H. Wasche 233 1. The Concept of a Vector Space 235 2. Linear Transformations of Vector Spaces 246 3. Products of Vectors 266 CHAPTER 4 Polynomials, G. Pickert and W. Ruckert 291 1. Entire Rational Functions 291 2. Polynomials 296 3. The Use of Indeterminates as a Method of Proof 312 CHAPTER 5 Rings and Ideals, W. Grobner and P. Lesky 316 1. Rings, Integral Domains, Fields 316 CONTENTS vii 2. Divisibility in Integral Domains 327 3. Ideals in Commutative Rings, Principal Ideal Rings, Residue Class Rings 338 4. Divisibility in Polynomial Rings Elimination 346 CHAPTER 6 Theory of Numbers, H.-H. Ostmann and H. Liermann 355 1. Introduction 355 2. Divisibility Theory 355 3. Continued Fractions 372 4. Congruences 380 5. Some Number-Theoretic Functions; The Mobius Inversion Formula 388 6. The Chinese Remainder Theorem; Direct Decomposition of (f,/{m) 391 7. Diophantine Equations; Algebraic Congruences 395 8. Algebraic Numbers 401 9. Additive Number Theory 405 CHAPTER 7 Algebraic Extensions of a Field, O. Haupt and P. Sengenhorst 409 1. The Splitting Field of a Polynomial 410 2. Finite Extensions 418 3. Normal Extensions 420 4. Separable Extensions 422 5. Roots of Unity 425 6. Isomorphic Mappings of Separable Finite Extensions 431 7. Normal Fields and the A utomorphism Group ( Galois Group) 433 8. Finite Fields 438 9. Irreducibility of the Cyclotomic Polynomial and Structure of the Galois Group of the Cyclotomic Field over the Field of Rational Numbers 448 10. Solvability by Radicals. Equations of the Third and Fourth Degree 452 CHAPTER 8 Complex Numbers and Quaternions, G. Pickert and H.-G. Steiner 456 1. The Complex Numbers 456 viii CONTENTS 2. Algebraic Closedness of the Field of Complex Numbers 462 3. Quaternions 467 CHAPTER 9 Lattices, H. Gericke and H. Martens 483 I. Properties of the Power Set 485 2. Examples 490 3. Lattices of Finite Length 495 4. Distributive Lattices 497 5. Modular Lattices 501 6. Projective Geometry 505 CHAPTER 10 Some Basic Concepts for a Theory of Structure, H. Gericke and H. Martens 508 I. Configurations 509 2. Structure 515 CHAPTER II Zorn's Lemma and the High Chain Principle, H. Wolff and H. Noack 522 I. Ordered Sets 522 2. Zorn's Lemma 524 3. Examples of the Application of Zorn's Lemma 525 4. Proof of Zorn's Lemma from the Axiom of Choice 529 5. Questions Concerning the Foundations of Mathematics 534 Bibliography 536 Index 537