Galois Theory

Contents......Page 009
Preface......Page 005
Notation......Page 019
PART I POLYNOMIALS......Page p1
1.1 Cardan\'s Formulas......Page p3
Historical Notes......Page p8
A Permutations......Page p10
B The Discriminant......Page p11
C Symmetric Polynomials......Page p13
Historical Notes......Page p14
A The Number of Real Roots......Page p15
B Trigonometric Solution of the Cubic......Page p18
Historical Notes......Page p19
References......Page p23
A The Polynomial Ring in n Variables......Page p25
B Elementary Symmetric Polynomials......Page p27
Mathematical Notes......Page p29
A The Fundamental Theorem......Page p30
B The Roots of a Polynomial......Page p34
C Uniqueness......Page p35
Historical Notes......Page p37
2.3 Computing w. Symm. Polynomials (Optional)......Page p41
A Using Mathematica......Page p42
B Using Maple......Page p43
2.4 The Discriminant......Page p46
Mathematical Notes......Page p48
Historical Notes......Page p50
References......Page p53
3.1 The Existence of Roots......Page p55
Mathematical Notes......Page p59
Historical Notes......Page p61
3.2 The Fundamental Theorem of Algebra......Page p62
Mathematical Notes......Page p66
Historical Notes......Page p67
References......Page p70
PART II FIELDS......Page p71
4.1 Elements of Extension Fields......Page p73
A Minimal Polynomials......Page p74
B Adjoining Elements......Page p75
Historical Notes......Page p79
A Using Maple and Mathematica......Page p81
B Algorithms for Factoring......Page p83
C Schonemann-Eisenstein Criterion......Page p84
D Prime Radicals......Page p85
Historical Notes......Page p87
4.3 The Degree of an Extension......Page p88
A Finite Extensions......Page p89
B The Tower Theorem......Page p91
Historical Notes......Page p93
4.4 Algebraic Extensions......Page p94
Mathematical Notes......Page p97
References......Page p98
A Definitions and Examples......Page p101
B Uniqueness......Page p103
5.2 Normal Extensions......Page p107
Historical Notes......Page p108
5.3 Separable Extensions......Page p109
A Fields of Characteristic 0......Page p112
B Fields of Characteristic p......Page p113
C Computations......Page p114
Mathematical Notes......Page p116
5.4 The Theorem of the Primitive Element......Page p119
Historical Notes......Page p122
References......Page p123
6.1 Definition of the Galois Group......Page p125
Historical Notes......Page p128
6.2 Galois Groups of Splitting Fields......Page p130
6.3 Permutations of the Roots......Page p132
Mathematical Notes......Page p134
Historical Notes......Page p135
A The pth Roots of 2......Page p136
B The Universal Extension......Page p137
C A Polynomial of Degree 5......Page p138
Mathematical Notes......Page p139
Historical Notes......Page p141
6.5 Abelian Equations (Optional)......Page p143
Historical Notes......Page p144
References......Page p146
A Splitting Fields of Separable Polynomials......Page p147
B Finite Separable Extensions......Page p150
C Galois Closures......Page p151
Historical Notes......Page p152
A Conjugate Fields......Page p154
B Normal Subgroups......Page p155
Mathematical Notes......Page p159
Historical Notes......Page p160
7.3 The Fundamental Theorem of Galois Theory......Page p161
A The Discriminant......Page p167
B The Universal Extension......Page p169
C The Inverse Galois Problem......Page p170
Historical Notes......Page p172
A Groups of Automorphisms......Page p173
B Function Fields in One Variable......Page p176
C Linear Fractional Transformations......Page p178
D Stereographic Projection......Page p180
Mathematical Notes......Page p183
References......Page p188
Part III APPLICATIONS......Page p189
8.1 Solvable Groups......Page p191
Mathematical Notes......Page p194
A Definitions and Examples......Page p196
C Properties of Radical & Solvable Extensions......Page p198
Historical Notes......Page p200
A Roots of Unity and Lagrange Resolvents......Page p201
B Galois\'s Theorem......Page p204
C Cardan\'s Formulas......Page p207
Historical Notes......Page p208
8.4 Simple Groups......Page p210
Historical Notes......Page p214
A Roots and Radicals......Page p215
C Abelian Equations......Page p217
D Fundamental Theorem of Algebra Revisited......Page p218
Historical Notes......Page p219
A Real Radicals......Page p220
B Irred. Polyns. with Real Radical Roots......Page p222
C Failure of Solvability in Char. p......Page p224
Historical Notes......Page p226
References......Page p227
A Some Number Theory......Page p229
B Definition of Cyclotomic Polynomials......Page p231
C Galois Group of a Cyclotomic Extension......Page p233
Historical Notes......Page p235
A The Galois Correspondence......Page p238
B Periods......Page p239
C Explicit Calculations......Page p242
D Solvability by Radicals......Page p246
Mathematical Notes......Page p248
Historical Notes......Page p249
References......Page p254
10.1 Constructible Numbers......Page p255
Mathematical Notes......Page p263
Historical Notes......Page p266
10.2 Regular Polygons and Roots of Unity......Page p269
Historical Notes......Page p271
10.3 0rigami (Optional)......Page p273
A Origami Constructions......Page p274
B Origami Numbers......Page p276
C Marked Rulers & Intersections of Conics......Page p279
Mathematical Notes......Page p281
Historical Notes......Page p282
References......Page p287
A Existence and Uniqueness......Page p289
B Galois Groups......Page p292
Mathematical Notes......Page p294
Historical Notes......Page p295
A Irreducible Polynomials of Fixed Degree......Page p299
B Cyclotomic Polynomials Modulo p......Page p301
C Berlekamp\'s Algorithm......Page p303
Historical Notes......Page p305
References......Page p308
PART IV FURTHER TOPICS......Page p311
12.1 Lagrange......Page p313
A Resolvent Polynomials......Page p314
B Similar Functions......Page p318
C The Quartic......Page p321
D Higher Degrees......Page p324
E Lagrange Resolvents......Page p326
Historical Notes......Page p327
12.2 Galois......Page p332
B Galois Resolvents......Page p333
C Galois\'s Group......Page p336
D Natural & Accessory Irrationalities......Page p337
E Galois\'s Strategy......Page p339
Historical Notes......Page p341
A Algebraic Quantities......Page p346
B Module Systems......Page p347
C Splitting Fields......Page p349
Historical Notes......Page p352
References......Page p354
13.1 Quartic Polynomials......Page p357
Mathematical Notes......Page p362
Historical Notes......Page p365
A Transitive Subgroups of S5......Page p367
B Galois Groups of Quintics......Page p370
C Examples......Page p375
D Solvable Quintics......Page p376
Mathematical Notes......Page p377
Historical Notes......Page p379
A Jordan\'s Strategy......Page p384
B Relative Resolvents......Page p388
C Factoring Resolvents......Page p389
Mathematical Notes......Page p391
A Kronecker\'s Analysis......Page p394
B Dedekind\'s Theorem......Page p398
Mathematical Notes......Page p401
References......Page p404
14.1 Polynomials of Prime Degree......Page p407
Mathematical Notes......Page p410
Historical Notes......Page p411
14.2 Imprimitive Polyns. of Prime-Squared Degree......Page p412
A Primitive and Imprimitive Groups......Page p413
B Wreath Products......Page p414
C The Solvable Case......Page p417
Historical Notes......Page p419
14.3 Primitive Permutation Groups......Page p422
A Doubly Transitive Permutation Groups......Page p423
B Affine Linear and Semilinear Groups......Page p424
C Minimal Normal Subgroups......Page p425
D The Solvable Case......Page p427
Mathematical Notes......Page p431
Historical Notes......Page p433
A The First Two Subgroups......Page p438
B The Third Subgroup......Page p439
C The Solvable Case......Page p444
Mathematical Notes......Page p451
Historical Notes......Page p452
References......Page p455
15 The Lemniscate......Page p457
A Division Points......Page p458
B Arc Length of the Lemniscate......Page p460
Mathematical Notes......Page p462
Historical Notes......Page p463
A A Periodic Function......Page p465
B Addition Laws......Page p467
C Multiplication by Integers......Page p470
Historical Notes......Page p473
15.3 The Complex Lemniscatic Function......Page p476
A A Doubly Periodic Function......Page p477
B Zeros and Poles......Page p479
Mathematical Notes......Page p481
Historical Notes......Page p482
15.4 Complex Multiplication......Page p483
B Multiplication by Gaussian Integers......Page p485
C Multiplication by Gaussian Primes......Page p492
Mathematical Notes......Page p495
Historical Notes......Page p496
A The Lemniscatic Galois Group......Page p499
B Straightedge-and-Compass Constructions......Page p500
Mathematical Notes......Page p503
Historical Notes......Page p504
References......Page p507
A Groups......Page p509
B Rings......Page p513
C Fields......Page p5]4
D Polynomials......Page p515
A Addition, Multiplication, and Division......Page p518
B Roots of Complex Numbers......Page p519
A.3 Polynomials with Rational Coefficients......Page p522
A.4 Group Actions......Page p523
A The Sylow Theorems......Page p526
C The Multiplicative Group of a Field......Page p527
D Unique Factorization Domains......Page p528
Appendix B Hints to Selected Exercises......Page p529
A Books and Monographs on Galois Theory......Page p543
C Collected Works......Page p544
Index......Page p547