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Galois theory

Author: David A Cox
Publisher: Hoboken, N.J. : John Wiley & Sons, ©2012.
Series: Pure and applied mathematics (John Wiley & Sons : Unnumbered)
Edition/Format:   Print book : English : 2nd edView all editions and formats

This book brings one of the most colorful and influential theories in algebra to life for professional algebraists and students alike. The Second Edition features new exercises and an updated  Read more...

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Document Type: Book
All Authors / Contributors: David A Cox
ISBN: 9781118072059 1118072057
OCLC Number: 755640849
Description: xxviii, 570 pages : illustrations ; 24 cm.
Contents: <p>Preface to the First Edition xvii <p>Preface to the Second Edition xxi <p>Notation xxiii <p>1 Basic Notation xxiii <p>2 Chapter-by-Chapter Notation xxv <p>PART I POLYNOMIALS <p>1 Cubic Equations 3 <p>1.1 Cardan's Formulas 4 <p>1.2 Permutations of the Roots 10 <p>1.3 Cubic Equations over the Real Numbers 15 <p>2 Symmetric Polynomials 25 <p>2.1 Polynomials of Several Variables 25 <p>2.2 Symmetric Polynomials 30 <p>2.3 Computing with Symmetric Polynomials (Optional) 42 <p>2.4 The Discriminant 46 <p>3 Roots of Polynomials 55 <p>3.1 The Existence of Roots 55 <p>3.2 The Fundamental Theorem of Algebra 62 <p>PART II FIELDS <p>4 Extension Fields 73 <p>4.1 Elements of Extension Fields 73 <p>4.2 Irreducible Polynomials 81 <p>4.3 The Degree of an Extension 89 <p>4.4 Algebraic Extensions 95 <p>5 Normal and Separable Extensions 101 <p>5.1 Splitting Fields 101 <p>5.2 Normal Extensions 107 <p>5.3 Separable Extensions 109 <p>5.4 Theorem of the Primitive Element 119 <p>6 The Galois Group 125 <p>6.1 Definition of the Galois Group 125 <p>6.2 Galois Groups of Splitting Fields 130 <p>6.3 Permutations of the Roots 132 <p>6.4 Examples of Galois Groups 136 <p>6.5 Abelian Equations (Optional) 143 <p>7 The Galois Correspondence 147 <p>7.1 Galois Extensions 147 <p>7.2 Normal Subgroups and Normal Extensions 154 <p>7.3 The Fundamental Theorem of Galois Theory 161 <p>7.4 First Applications 167 <p>7.5 Automorphisms and Geometry (Optional) 173 <p>PART III APPLICATIONS <p>8 Solvability by Radicals 191 <p>8.1 Solvable Groups 191 <p>8.2 Radical and Solvable Extensions 196 <p>8.3 Solvable Extensions and Solvable Groups 201 <p>8.4 Simple Groups 210 <p>8.5 Solving Polynomials by Radicals 215 <p>8.6 The Casus Irreducbilis (Optional) 220 <p>9 Cyclotomic Extensions 229 <p>9.1 Cyclotomic Polynomials 229 <p>9.2 Gauss and Roots of Unity (Optional) 238 <p>10 Geometric Constructions 255 <p>10.1 Constructible Numbers 255 <p>10.2 Regular Polygons and Roots of Unity 270 <p>10.3 Origami (Optional) 274 <p>11 Finite Fields 291 <p>11.1 The Structure of Finite Fields 291 <p>11.2 Irreducible Polynomials over Finite Fields (Optional) 301 <p>PART IV FURTHER TOPICS <p>12 Lagrange, Galois, and Kronecker 315 <p>12.1 Lagrange 315 <p>12.2 Galois 334 <p>12.3 Kronecker 347 <p>13 Computing Galois Groups 357 <p>13.1 Quartic Polynomials 357 <p>13.2 Quintic Polynomials 368 <p>13.3 Resolvents 386 <p>13.4 Other Methods 400 <p>14 Solvable Permutation Groups 413 <p>14.1 Polynomials of Prime Degree 413 <p>14.2 Imprimitive Polynomials of Prime-Squared Degree 419 <p>14.3 Primitive Permutation Groups 429 <p>14.4 Primitive Polynomials of Prime-Squared Degree 444 <p>15 The Lemniscate 463 <p>15.1 Division Points and Arc Length 464 <p>15.2 The Lemniscatic Function 470 <p>15.3 The Complex Lemniscatic Function 482 <p>15.4 Complex Multiplication 489 <p>15.5 Abel's Theorem 504 <p>A Abstract Algebra 515 <p>A.1 Basic Algebra 515 <p>A.2 Complex Numbers 524 <p>A.3 Polynomials with Rational Coefficients 528 <p>A.4 Group Actions 530 <p>A.5 More Algebra 532 <p>Index 557
Series Title: Pure and applied mathematics (John Wiley & Sons : Unnumbered)
Responsibility: David A. Cox.
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<p> There is barely a better introduction to the subject, inall its theoretical and practical aspects, than the book underreview. (Zentralblatt MATH, 1 December2012) <p> <p> <p>

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