Galois Theory

David A. Cox 지음

Wiley 2012년 03월 27일 출간

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Praise for the First Editionquot;. . .will certainly fascinate anyone interested in abstract algebra: a remarkable book!quot;m-Monatshefte fur MathematikGalois theory is one of the most established topics in mathematics, with historical roots that led to the development of many central concepts in modern algebra, including groups and fields. Covering classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields, Galois Theory, Second Edition delves into novel topics like Abel's theory of Abelian equations, casus irreducibili, and the Galois theory of origami.In addition, this book features detailed treatments of several topics not covered in standard texts on Galois theory, including:The contributions of Lagrange, Galois, and KroneckerHow to compute Galois groupsGalois's results about irreducible polynomials of prime orprime-squared degreeAbel's theorem about geometric constructions on the lemniscatesGalois groups of quartic polynomials in all characteristicsThroughout the book, intriguing Mathematical Notes and Historical Notes sections clarify the discussed ideas and the historical context; numerous exercises and examples use Maple and Mathematica to showcase the computations related to Gal

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Preface to the First Edition xviiPreface to the Second Edition xxiNotation xxiii1 Basic Notation xxiii2 Chapter-by-Chapter Notation xxvPART I POLYNOMIALS1 Cubic Equations 31.1 Cardan's Formulas 41.2 Permutations of the Roots 101.3 Cubic Equations over the Real Numbers 152 Symmetric Polynomials 252.1 Polynomials of Several Variables 252.2 Symmetric Polynomials 302.3 Computing with Symmetric Polynomials (Optional) 422.4 The Discriminant 463 Roots of Polynomials 553.1 The Existence of Roots 553.2 The Fundamental Theorem of Algebra 62PART II FIELDS4 Extension Fields 734.1 Elements of Extension Fields 734.2 Irreducible Polynomials 814.3 The Degree of an Extension 894.4 Algebraic Extensions 955 Normal and Separable Extensions 1015.1 Splitting Fields 1015.2 Normal Extensions 1075.3 Separable Extensions 1095.4 Theorem of the Primitive Element 1196 The Galois Group 1256.1 Definition of the Galois Group 1256.2 Galois Groups of Splitting Fields 1306.3 Permutations of the Roots 1326.4 Examples of Galois Groups 1366.5 Abelian Equations (Optional) 1437 The Galois Correspondence 1477.1 Galois Extensions 1477.2 Normal Subgroups and Normal Extensions 1547.3 The Fundamental Theorem of Galois Theory 1617.4 First Applications 1677.5 Automorphisms and Geometry (Optional) 173PART III APPLICATIONS8 Solvability by Radicals 1918.1 Solvable Groups 1918.2 Radical and Solvable Extensions 1968.3 Solvable Extensions and Solvable Groups 2018.4 Simple Groups 2108.5 Solving Polynomials by Radicals 2158.6 The Casus Irreducbilis (Optional) 2209 Cyclotomic Extensions 2299.

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“There is barely a better introduction to the subject, in all its theoretical and practical aspects, than the book under review.”& (Zentralblatt MATH, 1 December 2012)&&&

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저자(글) David A. Cox

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DAVID A. COX, PhD, is Professor in the Department of Mathematics at Amherst College. He has published extensively in his areas of research interest, which include algebraic geometry, number theory, and the history of mathematics. Dr. Cox is consulting editor for Wiley's Pure and Applied Mathematics book series and the author of Primes of the Form x2 + ny2 (Wiley).