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[资源] Field Theory - Steven Roman

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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