.

0 (hodnocen0 x )

BK

New York : Springer, 1995

xii,272 s.

ISBN 0-387-94408-7 (brož.)

Bibliogr. s. 265-266.

000031016

// Sections marked with an asterisk are optional. // Preface // Chapter 0 // Preliminaries // Lattices // Grou // Rings // Integral Domains // Unique Factorization Domains // Principal Ideal Domains // Euclidean Domains // Tensor Products // Part 1 Basic Theory // Chapter 1 // Polynomials 25 // Polynomials Over a Ring // itive Polynomials // The Division Algorithm // Splitting Fields // The Minimal Polynomial // Multiple Roots // Testing for Irreducibility // Chapter 2 // Field Extensions // 2.1 The Lattice of Subfields of a Field // 2.2 Distinguished Extensions // 2.3 Finitely Generated Extensions // 2.4 Simple Extensions // 2.5 Finite Extensions // 2.6 Algebraic Extensions // 2.7 Algebraic Closures // 2.8 Embeddings // 2.9 Splitting Fields and Normal Extensions // Chapter 3 // Algebraic Independence // 3.1 Dependence Relations // 3.2 Algebraic Dependence // 3.3 Transcendence Bases // 3.4 Simple Transcendental Extensions // Chapter 4 // Separability // 4.1 Separable Polynomials // 4.2 Separable Degree // 4.3 The Simple Case // 4.4 The Finite Case // 4.5 The Algebraic Case // 4.6 Pure Inseparability // 4.7 Separable and Purely Inseparable Closures // 4.8 Perfect Fields // Part 2 Galois Theory // Chapter 5 // Galois Theory I // 5.1 Galois Connections // 5.2 The Galois Correspondence // 5.3 Who’s Closed? // Normal Subgroup and Normal Extensions // More on Galois Groups // 5.6 Linear Disjointness // 5.7 The Krull Topology 120 // Chapter 6 // Galois Theory II // 6.1 The Galois Group of a Polynomial // 6.2 Symmetric Polynomials // 6.3 The Discriminant of a Polynomial // The Galois Groups of Some Sj all Degree Polynomials // Chapter 7 // A Field Extension as a Vector Space // 7.1 The Norm and the Trace // 7.2 The Discriminant of Field Elements // 7.3 Algebraic Independence of Embeddings // 7.4 The Normal Basis Theorem // Chapter 8 // Finite Fields I: Basic Properties //

8.1 Finite Fields 161 // 8.2 Finite Fields as Splitting Fields 162 // 8.3 The Subfields of a Finite Field 163 // 8.4 The Multiplicative Structure of a Finite Field 163 // 8.5 The Galois Group of a Finite Field 165 // 8.6 Irreducible Polynomials over Finite Fields 165 // 8.7 Normal Bases 169 // 8.8 The Algebraic Closure of a Finite Field 170 // Chapter 9 // Finite Fields II: Additional Properties 175 // 9.1 Finite Field Arithmetic // 9.2 The Number of Irreducible Polynomials // 9.3 Polynomial Functions // 9.4 Linearized Polynomials // Part 3 The Theory of Binomials 187 // Chapter 10 // The Roots of Unity 189 // 10.1 Roots of Unity // Cyclotomic Extensions // Nor // al Bases and Roots of Unity // Wedderburn’s Theorem 200 // 10.5 Realizing Groups as Galois Group 201 // Chapter 11 // Cyclic Extensions 209 // 11.1 Cyclic Extensions 210 // 11.2 Extensions of Degree Char(F) 212 // Chapter 12 // Solvable Extensions 215 // 12.1 Solvable Groups 215 // 12.2 Solvable Extensions 216 // 12.3 Solvability by Radicals 219 // 12.4 Polynomial Equations 222 // Chapter 13 // Binomials 227 // 13.1 Irreducibility 228 // 13.2 The Galois Group of a Binomial 232 // 13.3 The Independence of Irrational Numbers 241 // Chapter 14 // Families of Binomials 247 // 14.1 The Splitting Field // er Theory 249 // Appendix // Möbius Inversion 257 // References // Index of Symbols // Index