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Field Theory - Steven Roman
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Field Theory - Steven Roman Preface.................................................................................................... vii Contents...................................................................................................ix 0 Preliminaries...................................................................................1 0.1 Lattices..................................................................................................1 0.2 Groups.................................................................................................. 2 0.3 The Symmetric Group........................................................................ 10 0.4 Rings...................................................................................................10 0.5 Integral Domains................................................................................ 14 0.6 Unique Factorization Domains........................................................... 16 0.7 Principal Ideal Domains..................................................................... 16 0.8 Euclidean Domains.............................................................................17 0.9 Tensor Products.................................................................................. 17 Exercises...................................................................................................19 Part I—Field Extensions 1 Polynomials...................................................................................23 1.1 Polynomials over a Ring.....................................................................23 1.2 Primitive Polynomials and Irreducibility............................................24 1.3 The Division Algorithm and Its Consequences.................................. 27 1.4 Splitting Fields....................................................................................32 1.5 The Minimal Polynomial....................................................................32 1.6 Multiple Roots.................................................................................... 33 1.7 Testing for Irreducibility.....................................................................35 Exercises...................................................................................................38 2 Field Extensions............................................................................41 2.1 The Lattice of Subfields of a Field..................................................... 41 2.2 Types of Field Extensions.................................................................. 42 2.3 Finitely Generated Extensions............................................................46 2.4 Simple Extensions.............................................................................. 47 2.5 Finite Extensions................................................................................ 53 2.6 Algebraic Extensions..........................................................................54 x Contents 2.7 Algebraic Closures............................................................................. 56 2.8 Embeddings and Their Extensions..................................................... 58 2.9 Splitting Fields and Normal Extensions............................................. 63 Exercises...................................................................................................66 3 Embeddings and Separability..................................................... 73 3.1 Recap and a Useful Lemma................................................................73 3.2 The Number of Extensions: Separable Degree...................................75 3.3 Separable Extensions..........................................................................77 3.4 Perfect Fields...................................................................................... 84 3.5 Pure Inseparability..............................................................................85 *3.6 Separable and Purely Inseparable Closures......................................88 Exercises...................................................................................................91 4 Algebraic Independence...............................................................93 4.1 Dependence Relations........................................................................ 93 4.2 Algebraic Dependence........................................................................96 4.3 Transcendence Bases........................................................................100 *4.4 Simple Transcendental Extensions................................................. 105 Exercises.................................................................................................108 Part II—Galois Theory 5 Galois Theory I: An Historical Perspective............................. 113 5.1 The Quadratic Equation....................................................................113 5.2 The Cubic and Quartic Equations.....................................................114 5.3 Higher-Degree Equations................................................................. 116 5.4 Newton's Contribution: Symmetric Polynomials..............................117 5.5 Vandermonde....................................................................................119 5.6 Lagrange........................................................................................... 121 5.7 Gauss................................................................................................ 124 5.8 Back to Lagrange..............................................................................128 5.9 Galois................................................................................................130 5.10 A Very Brief Look at the Life of Galois.........................................135 6 Galois Theory II: The Theory................................................... 137 6.1 Galois Connections...........................................................................137 6.2 The Galois Correspondence..............................................................143 6.3 Who's Closed?.................................................................................. 148 6.4 Normal Subgroups and Normal Extensions......................................154 6.5 More on Galois Groups.................................................................... 159 6.6 Abelian and Cyclic Extensions.........................................................164 *6.7 Linear Disjointness......................................................................... 165 Exercises.................................................................................................168 7 Galois Theory III: The Galois Group of a Polynomial........... 173 7.1 The Galois Group of a Polynomial...................................................173 7.2 Symmetric Polynomials....................................................................174 7.3 The Fundamental Theorem of Algebra.............................................179 Contents xi 7.4 The Discriminant of a Polynomial....................................................180 7.5 The Galois Groups of Some Small-Degree Polynomials..................182 Exercises.................................................................................................193 8 A Field Extension as a Vector Space........................................ 197 8.1 The Norm and the Trace...................................................................197 *8.2 Characterizing Bases...................................................................... 202 *8.3 The Normal Basis Theorem............................................................206 Exercises.................................................................................................208 9 Finite Fields I: Basic Properties................................................ 211 9.1 Finite Fields Redux...........................................................................211 9.2 Finite Fields as Splitting Fields........................................................ 212 9.3 The Subfields of a Finite Field......................................................... 213 9.4 The Multiplicative Structure of a Finite Field.................................. 214 9.5 The Galois Group of a Finite Field...................................................215 9.6 Irreducible Polynomials over Finite Fields.......................................215 *9.7 Normal Bases..................................................................................218 *9.8 The Algebraic Closure of a Finite Field......................................... 219 Exercises.................................................................................................223 10 Finite Fields II: Additional Properties..................................... 225 10.1 Finite Field Arithmetic................................................................... 225 *10.2 The Number of Irreducible Polynomials...................................... 232 *10.3 Polynomial Functions................................................................... 234 *10.4 Linearized Polynomials................................................................ 236 Exercises.................................................................................................238 11 The Roots of Unity......................................................................239 11.1 Roots of Unity................................................................................ 239 11.2 Cyclotomic Extensions................................................................... 241 *11.3 Normal Bases and Roots of Unity................................................ 250 *11.4 Wedderburn's Theorem.................................................................251 *11.5 Realizing Groups as Galois Groups..............................................253 Exercises.................................................................................................257 12 Cyclic Extensions........................................................................261 12.1 Cyclic Extensions........................................................................... 261 12.2 Extensions of Degree Char􀂲􀀭 􀂳.......................................................265 Exercises.................................................................................................266 13 Solvable Extensions....................................................................269 13.1 Solvable Groups............................................................................. 269 13.2 Solvable Extensions........................................................................270 13.3 Radical Extensions......................................................................... 273 13.4 Solvability by Radicals................................................................... 274 13.5 Solvable Equivalent to Solvable by Radicals................................. 276 13.6 Natural and Accessory Irrationalities............................................. 278 13.7 Polynomial Equations.....................................................................280 xii Contents Exercises.................................................................................................282 Part III—The Theory of Binomials 14 Binomials.....................................................................................289 14.1 Irreducibility................................................................................... 289 14.2 The Galois Group of a Binomial.................................................... 296 *14.3 The Independence of Irrational Numbers..................................... 304 Exercises.................................................................................................307 15 Families of Binomials................................................................. 309 15.1 The Splitting Field.......................................................................... 309 15.2 Dual Groups and Pairings...............................................................310 15.3 Kummer Theory..............................................................................312 Exercises.................................................................................................316 Appendix: Möbius Inversion..............................................................319 Partially Ordered Sets.............................................................................319 The Incidence Algebra of a Partially Ordered Set..................................320 Classical Mo¨bius Inversion.....................................................................324 Multiplicative Version of Mo¨bius Inversion.......................................... 325 References............................................................................................ 327 Index..................................................................................................... 329 |
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