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Edition: 98

Copyright: 1998

Publisher: Springer-Verlag New York

Published: 1998

International: No

Copyright: 1998

Publisher: Springer-Verlag New York

Published: 1998

International: No

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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.

Symmetry. Rings.

Domains and Fields.

Homomorphisms and Ideals.

Quotient Rings.

Polynomial Rings over Fields.

Prime Ideals and Maximal Ideals.

Irreducible Polynomials.

Classical Formulas.

Splitting Fields.

The Galois Group.

Roots of Unity.

Solvability by Radicals.

Independence of Characters.

Galois Extensions.

The Fundamental

Theorem of Galois Theory.

Applications.

Galois's Great Theorem. Discriminants.

Galois Groups of Quadratics, Dubics, and Quartics.

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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.

Table of Contents

Symmetry. Rings.

Domains and Fields.

Homomorphisms and Ideals.

Quotient Rings.

Polynomial Rings over Fields.

Prime Ideals and Maximal Ideals.

Irreducible Polynomials.

Classical Formulas.

Splitting Fields.

The Galois Group.

Roots of Unity.

Solvability by Radicals.

Independence of Characters.

Galois Extensions.

The Fundamental

Theorem of Galois Theory.

Applications.

Galois's Great Theorem. Discriminants.

Galois Groups of Quadratics, Dubics, and Quartics.

Publisher Info

Publisher: Springer-Verlag New York

Published: 1998

International: No

Published: 1998

International: No

Domains and Fields.

Homomorphisms and Ideals.

Quotient Rings.

Polynomial Rings over Fields.

Prime Ideals and Maximal Ideals.

Irreducible Polynomials.

Classical Formulas.

Splitting Fields.

The Galois Group.

Roots of Unity.

Solvability by Radicals.

Independence of Characters.

Galois Extensions.

The Fundamental

Theorem of Galois Theory.

Applications.

Galois's Great Theorem. Discriminants.

Galois Groups of Quadratics, Dubics, and Quartics.