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Galois Theory (Universitext)by Joseph Rotman
Synopses & ReviewsPublisher Comments: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, rulercompass constructions, and the early history of Galois Theory. This book provides a concise introduction to Galois Theory suitable for firstyear 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).
Synopsis:"The exposition is quite efficient, packing much material in a limited number of pages. . . . The greatest asset of this book is its nice selection of topics, focusing on the fundamental theorem of Galois theory and its application to solvability of equations by radicals. . . The style is nononsense, crisp . . "."The American Mathematical Monthly" (on first edition).
Synopsis:A clear, efficient exposition of this topic with complete proofs and exercises, covering cubic and quartic formulas; fundamental theory of Galois theory; insolvability of the quintic; Galoiss Great Theorem; and computation of Galois groups of cubics and quartics. Suitable for firstyear 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. It 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  an analogy which serves to help readers organise the various field theoretic definitions and constructions. The text is rounded off by appendices on group theory, rulercompass constructions, and the early history of Galois Theory. The exposition has been redesigned 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 ContentsSymmetry. 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. Epilogue. Appendices.
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