Galois Theory

by Rotman, Joseph J.

• ISBN13: 9780387985411
• ISBN10: 0387985417
• Edition: 2nd
• Format: Paperback
• Copyright: 1998-10-01
• Publisher: Springer Verlag

Summary

This text offers a clear, efficient exposition of Galois Theory with complete proofs and exercises. Topics include: cubic and quartic formulas; Fundamental Theory of Galois Theory; insolvability of the quintic; Galois's Great Theorem (solvability by radicals of a polynomial is equivalent to solvability of its Galois Group); and computation of Galois groups of cubics and quartics. There are appendices on group theory, ruler-compass constructions, and the early history of Galois Theory. This book provides a concise introduction to Galois Theory suitable for first-year graduate students, either as a text for a course or for study outside the classroom.This new edition has been completely rewritten in an attempt to make proofs clearer by providing more details. The book now begins with a short section on symmetry groups of polygons in the plane, for there is an analogy between polygons and their symmetry groups and polynomials and their Galois groups; this analogy can serve as a guide by helping readers organize the various field theoretic definitions and constructions. The exposition has been reorganized so that the discussion of solvability by radicals now appears later and several new theorems not found in the first edition are included (e.g., Casus Irreducibilis).

Table of Contents

Preface to the Second Edition	vii
Preface to the First Edition	ix
To the Reader	xi
Symmetry	1	(6)
Rings	7	(6)
Domains and Fields	13	(4)
Homomorphisms and Ideals	17	(4)
Quotient Rings	21	(3)
Polynomial Rings over Fields	24	(7)
Prime Ideals and Maximal Ideals	31	(7)
Irreducible Polynomials	38	(6)
Classical Formulas	44	(6)
Splitting Fields	50	(9)
The Galois Group	59	(4)
Roots of Unity	63	(8)
Solvability by Radicals	71	(5)
Independence of Characters	76	(3)
Galois Extensions	79	(4)
The Fundamental Theorem of Galois Theory	83	(2)
Applications	85	(5)
Galois's Great Theorem	90	(5)
Discriminants	95	(5)
Galois Groups of Quadratics, Cubics, and Quartics	100	(7)
Epilogue	107	(2)
Appendix A: Group Theory Dictionary	109	(3)
Appendix B: Group Theory Used in the Text	112	(17)
Appendix C: Ruler-Compass Constructions	129	(9)
Appendix D: Old-fashioned Galois Theory	138	(13)
References	151	(2)
Index	153