Galois theory / David A. Cox.

By: Cox, David AMaterial type: TextTextSeries: Pure and applied mathematics (John Wiley & Sons : Unnumbered)Publication details: Hoboken, NJ : John Wiley & Sons, �2012Edition: 2nd edDescription: 1 online resource (xxviii, 570 pages) : illustrationsContent type: text Media type: computer Carrier type: online resourceISBN: 9781118218457; 1118218450; 9781118218426; 1118218426Subject(s): Galois theory | Galois theory | Mathematics | MATHEMATICS -- Algebra -- Intermediate | Galois theoryGenre/Form: Electronic books.Additional physical formats: Print version:: Galois theory.DDC classification: 512/.32 LOC classification: QA214 | .C69 2012ebOnline resources: Click here to access online
Contents:
Galois Theory; CONTENTS; Preface to the First Edition; Preface to the Second Edition; Notation; 1 Basic Notation; 2 Chapter-by-Chapter Notation; PART I POLYNOMIALS; 1 Cubic Equations; 1.1 Cardan's Formulas; Historical Notes; 1.2 Permutations of the Roots; A Permutations; B The Discriminant; C Symmetric Polynomials; Mathematical Notes; Historical Notes; 1.3 Cubic Equations over the Real Numbers; A The Number of Real Roots; B Trigonometric Solution of the Cubic; Historical Notes; References; 2 Symmetric Polynomials; 2.1 Polynomials of Several Variables; A The Polynomial Ring in n Variables.
B The Elementary Symmetric PolynomialsMathematical Notes; 2.2 Symmetric Polynomials; A The Fundamental Theorem; B The Roots of a Polynomial; C Uniqueness; Mathematical Notes; Historical Notes; 2.3 Computing with Symmetric Polynomials (Optional); A Using Mathematica; B Using Maple; 2.4 The Discriminant; Mathematical Notes; Historical Notes; References; 3 Roots of Polynomials; 3.1 The Existence of Roots; Mathematical Notes; Historical Notes; 3.2 The Fundamental Theorem of Algebra; Mathematical Notes; Historical Notes; References; PART II FIELDS; 4 Extension Fields.
4.1 Elements of Extension FieldsA Minimal Polynomials; B Adjoining Elements; Mathematical Notes; Historical Notes; 4.2 Irreducible Polynomials; A Using Maple and Mathematica; B Algorithms for Factoring; C The Sch�onemann-Eisenstein Criterion; D Prime Radicals; Historical Notes; 4.3 The Degree of an Extension; A Finite Extensions; B The Tower Theorem; Mathematical Notes; Historical Notes; 4.4 Algebraic Extensions; Mathematical Notes; References; 5 Normal and Separable Extensions; 5.1 Splitting Fields; A Definition and Examples; B Uniqueness; 5.2 Normal Extensions; Historical Notes.
5.3 Separable ExtensionsA Fields of Characteristic 0; B Fields of Characteristic p; C Computations; Mathematical Notes; 5.4 Theorem of the Primitive Element; Mathematical Notes; Historical Notes; References; 6 The Galois Group; 6.1 Definition of the Galois Group; Historical Notes; 6.2 Galois Groups of Splitting Fields; 6.3 Permutations of the Roots; Mathematical Notes; Historical Notes; 6.4 Examples of Galois Groups; A The pth Roots of 2; B The Universal Extension; C A Polynomial of Degree 5; Mathematical Notes; Historical Notes; 6.5 Abelian Equations (Optional); Historical Notes; References.
7 The Galois Correspondence7.1 Galois Extensions; A Splitting Fields of Separable Polynomials; B Finite Separable Extensions; C Galois Closures; Historical Notes; 7.2 Normal Subgroups and Normal Extensions; A Conjugate Fields; B Normal Subgroups; Mathematical Notes; Historical Notes; 7.3 The Fundamental Theorem of Galois Theory; 7.4 First Applications; A The Discriminant; B The Universal Extension; C The Inverse Galois Problem; Historical Notes; 7.5 Automorphisms and Geometry (Optional); A Groups of Automorphisms; B Function Fields in One Variable; C Linear Fractional Transformations.
Summary: Praise for the First Edition". . .will certainly fascinate anyone interested in abstract algebra: a remarkable book!" & mdash;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 & rsquo;s theory of Abelian equations, casus irreducibili, and the Galois.
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Includes bibliographical references and index.

Print version record.

Galois Theory; CONTENTS; Preface to the First Edition; Preface to the Second Edition; Notation; 1 Basic Notation; 2 Chapter-by-Chapter Notation; PART I POLYNOMIALS; 1 Cubic Equations; 1.1 Cardan's Formulas; Historical Notes; 1.2 Permutations of the Roots; A Permutations; B The Discriminant; C Symmetric Polynomials; Mathematical Notes; Historical Notes; 1.3 Cubic Equations over the Real Numbers; A The Number of Real Roots; B Trigonometric Solution of the Cubic; Historical Notes; References; 2 Symmetric Polynomials; 2.1 Polynomials of Several Variables; A The Polynomial Ring in n Variables.

B The Elementary Symmetric PolynomialsMathematical Notes; 2.2 Symmetric Polynomials; A The Fundamental Theorem; B The Roots of a Polynomial; C Uniqueness; Mathematical Notes; Historical Notes; 2.3 Computing with Symmetric Polynomials (Optional); A Using Mathematica; B Using Maple; 2.4 The Discriminant; Mathematical Notes; Historical Notes; References; 3 Roots of Polynomials; 3.1 The Existence of Roots; Mathematical Notes; Historical Notes; 3.2 The Fundamental Theorem of Algebra; Mathematical Notes; Historical Notes; References; PART II FIELDS; 4 Extension Fields.

4.1 Elements of Extension FieldsA Minimal Polynomials; B Adjoining Elements; Mathematical Notes; Historical Notes; 4.2 Irreducible Polynomials; A Using Maple and Mathematica; B Algorithms for Factoring; C The Sch�onemann-Eisenstein Criterion; D Prime Radicals; Historical Notes; 4.3 The Degree of an Extension; A Finite Extensions; B The Tower Theorem; Mathematical Notes; Historical Notes; 4.4 Algebraic Extensions; Mathematical Notes; References; 5 Normal and Separable Extensions; 5.1 Splitting Fields; A Definition and Examples; B Uniqueness; 5.2 Normal Extensions; Historical Notes.

5.3 Separable ExtensionsA Fields of Characteristic 0; B Fields of Characteristic p; C Computations; Mathematical Notes; 5.4 Theorem of the Primitive Element; Mathematical Notes; Historical Notes; References; 6 The Galois Group; 6.1 Definition of the Galois Group; Historical Notes; 6.2 Galois Groups of Splitting Fields; 6.3 Permutations of the Roots; Mathematical Notes; Historical Notes; 6.4 Examples of Galois Groups; A The pth Roots of 2; B The Universal Extension; C A Polynomial of Degree 5; Mathematical Notes; Historical Notes; 6.5 Abelian Equations (Optional); Historical Notes; References.

7 The Galois Correspondence7.1 Galois Extensions; A Splitting Fields of Separable Polynomials; B Finite Separable Extensions; C Galois Closures; Historical Notes; 7.2 Normal Subgroups and Normal Extensions; A Conjugate Fields; B Normal Subgroups; Mathematical Notes; Historical Notes; 7.3 The Fundamental Theorem of Galois Theory; 7.4 First Applications; A The Discriminant; B The Universal Extension; C The Inverse Galois Problem; Historical Notes; 7.5 Automorphisms and Geometry (Optional); A Groups of Automorphisms; B Function Fields in One Variable; C Linear Fractional Transformations.

Praise for the First Edition". . .will certainly fascinate anyone interested in abstract algebra: a remarkable book!" & mdash;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 & rsquo;s theory of Abelian equations, casus irreducibili, and the Galois.

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