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## Details

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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