Basic Algebra II

Title page......Page 1
Dedication and date-line......Page 2
Contents......Page 3
Contents of Basic Algebra I......Page 8
INTRODUCTION: CONCEPTS FROM SET THEORY. THE INTEGERS......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#19
0.1 The power set of a set......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#21
0.2 The Cartesian product set. Maps......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#22
0.3 Equivalence relations. Factoring a map through an equivalence relation......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#28
0.4 The natural numbers......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#33
0.5 The number system $\mathbb{Z}$ of integers......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#37
0.6 Some basic arithmetic facts about $\mathbb{Z}$......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#40
0.7 A word on cardinal numbers......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#42
1 MONOIDS AND GROUPS......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#44
1.1 Monoids of transformations and abstract monoids......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#46
1.2 Groups of transformations and abstract groups......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#49
1.3 Isomorphism. Cayley's theorem......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#54
1.4 Generalized associativity. Commutativity......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#57
1.5 Submonoids and subgroups generated by a subset. Cyclic groups......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#60
1.6 Cycle decomposition of permutations......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#66
1.7 Orbits. Cosets of a subgroup......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#69
1.8 Congruences. Quotient monoids and groups......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#72
1.9 Homomorphisms......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#76
1.10 Subgroups of a homomorphic image. Two basic isomorphism theorems......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#82
1.11 Free objects. Generators and relations......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#85
1.12 Groups acting on sets......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#89
1.13 Sylow's theorems......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#97
2 RINGS......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#103
2.1 Definition and elementary properties......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#104
2.2 Types of rings......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#108
2.3 Matrix rings......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#110
2.4 Quaternions......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#116
2.5 Ideals' quotient rings......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#119
2.6 Ideals and quotient rings for $\mathbb{Z}$......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#121
2.7 Homomorphisms of rings. Basic theorems......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#124
2.8 Anti-isomorphisms......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#129
2.9 Field of fractions of a commutative domain......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#133
2.10 Polynomial rings......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#137
2.11 Some properties of polynomial rings and applications......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#145
2.12 Polynomial functions......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#152
2.13 Symmetric polynomials......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#156
2.14 Factorial monoids and rings......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#158
2.15 Principal ideal domains and Euclidean domains......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#165
2.16 Polynomial extensions of factorial domains......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#169
2.17 "Rings" (rings without unit)......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#173
3 MODULES OVER A PRINCIPAL IDEAL DOMAIN......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#175
3.1 Ring of endomorphisms of an abelian group......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#176
3.2 Left and right modules......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#181
3.3 Fundamental concepts and results......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#184
3.4 Free modules and matrices......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#188
3.5 Direct sums of modules......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#193
3.6 Finitely generated modules over a p.i.d. Preliminary results......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#197
3.7 Equivalence of matrices with entries in a p.i.d......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#199
3.8 Structure theorem for finitely generated modules over a p.i.d......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#205
3.9 Torsion modules' primary components' invariance theorem......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#207
3.10 Applications to abelian groups and to linear transformations......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#212
3.11 The ring of endomorphisms of a finitely generated module over a p.i.d......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#222
4 GALOIS THEORY OF EQUATIONS......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#228
4.1 Preliminary results' some old' some new......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#231
4.2 Construction with straight-edge and compass......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#234
4.3 Splitting field of a polynomial......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#242
4.4 Multiple roots......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#247
4.5 The Galois group. The fundamental Galois pairing......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#252
4.6 Some results on finite groups......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#262
4.7 Galois' criterion for solvability by radicals......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#269
4.8 The Galois group as permutation group of the roots......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#274
4.9 The general equation of the nth degree......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#280
4.10 Equations with rational coefficients and symmetric group as Galois group......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#285
4.11 Constructible regular $n$-gons......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#289
4.12 Transcendence of $e$ and $\pi$. The Lindemann-Weierstrass theorem......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#295
4.13 Finite fields......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#305
4.14 Special bases for finite dimensional extension fields......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#308
4.15 Traces and norms......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#314
4.16 Mod $p$ reduction......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#319
5 REAL POLYNOMIAL EQUATIONS AND INEQUALITIES......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#324
5.1 Ordered fields. Real closed fields......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#325
5.2 Sturm's theorem......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#329
5.3 Formalized Euclidean algorithm and Sturm's theorem......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#334
5.4 Elimination procedures. Resultants......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#340
5.5 Decision method for an algebraic curve......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#345
5.6 Generalized Sturm's theorem. Tarski's principle......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#353
6 METRIC VECTOR SPACES AND THE CLASSICAL GROUPS......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#360
6.1 Linear functions and bilinear forms......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#361
6.2 Alternate forms......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#367
6.3 Quadratic forms and symmetric bilinear forms......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#372
6.4 Basic concepts of orthogonal geometry......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#379
6.5 Witt's cancellation theorem......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#385
6.6 The theorem of Cartan-Dieudonne......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#389
6.7 Structure of the linear group $GL_n(F)$......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#393
6.8 Structure of orthogonal groups......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#400
6.9 Symplectic geometry. The symplectic group......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#409
6.10 Orders of orthogonal and symplectic groups over a finite field......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#416
6.11 Postscript on hermitian forms and unitary geometry......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#419
7 ALGEBRAS OVER A FIELD......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#423
7.1 Definition and examples of associative algebras......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#424
7.2 Exterior algebras. Application to determinants......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#429
7.3 Regular matrix representations of associative algebras. Norms and traces......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#440
7.4 Change of base field. Transitivity of trace and norm......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#444
7.5 Non-associative algebras. Lie and Jordan algebras......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#448
7.6 Hurwitz' problem. Composition algebras......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#456
7.7 Frobenius' and Wedderburn's theorems on associative division algebras......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#469
8 LATTICES AND BOOLEAN ALGEBRAS......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#473
8.1 Partially ordered sets and lattices......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#474
8.2 Distributivity and modularity......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#479
8.3 The theorem of Jordan-Hoelder-Dedekind......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#484
8.4 The lattice of subspaces of a vector space. Fundamental theorem of projective geometry......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#486
8.5 Boolean algebras......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#492
8.6 The Moebius function of a partialy ordered set......Page Jacobson N. Basic algebra 1 (2ed., Freeman, 1985)(ISBN 0716714809)(K)(T)(O)(517s)_MAt_.djvu#498
Preface......Page 13
Preface to the First Edition......Page 15
INTRODUCTION......Page 17
0.1 Zorn's lemma......Page 18
0.2 Arithmetic of cardinal numbers......Page 19
0.3 Ordinal and cardinal numbers......Page 20
0.4 Sets and classes......Page 22
References......Page 23
1 CATEGORIES......Page 24
1.1 Definition and examples of categories......Page 25
1.2 Some basic categorical concepts......Page 31
1.3 Functors and natural transformations......Page 34
1.4 Equivalence of categories......Page 42
1.5 Products and coproducts......Page 48
1.6 The horn functors. Representable functors......Page 53
1.7 Universals......Page 56
1.8 Adjoints......Page 61
References......Page 67
2 UNIVERSAL ALGEBRA......Page 68
2.1 $\Omega$-algebras......Page 69
2.2 Subalgebras and products......Page 74
2.3 Homomorphisms and congruences......Page 76
2.4 The lattice of congruences. Subdirect products......Page 82
2.5 Direct and inverse limits......Page 86
2.6 Ultraproducts......Page 91
2.7 Free $\Omega$-algebras......Page 94
2.8 Varieties......Page 97
2.9 Free products of groups......Page 103
2.10 Internal characterization of varieties......Page 107
References......Page 109
3 MODULES......Page 110
3.1 The categories R-mod and mod-R......Page 111
3.2 Artinian and Noetherian modules......Page 116
3.3 Schreier refinement theorem. Jordan-Hoelder theorem......Page 120
3.4 The Krull-Schmidt theorem......Page 126
3.5 Completely reducible modules......Page 133
3.6 Abstract dependence relations. Invariance of dimensionality......Page 138
3.7 Tensor products of modules......Page 141
3.8 Bimodules......Page 149
3.9 Algebras and coalgebras......Page 153
3.10 Projective modules......Page 164
3.11 Injective modules. Injective hull......Page 172
3.12 Morita contexts......Page 180
3.13 The Wedderburn-Artin theorem for simple rings......Page 187
3.14 Generators and progenerators......Page 189
3.15 Equivalence of categories of modules......Page 193
References......Page 199
4 BASIC STRUCTURE THEORY OF RINGS......Page 200
4.1 Primitivity and semi-primitivity......Page 201
4.2 The radical of a ring......Page 208
4.3 Density theorems......Page 213
4.4 Artinian rings......Page 218
4.5 Structure theory of algebras......Page 226
4.6 Finite dimensional central simple algebras......Page 231
4.7 The Brauer group......Page 242
4.8 Clifford algebras......Page 244
References......Page 261
5 CLASSICAL REPRESENTATION THEORY OF FINITE GROUPS......Page 262
5.1 Representations and matrix representations of groups......Page 263
5.2 Complete reducibility......Page 267
5.3 Application of the representation theory of algebras......Page 273
5.4 Irreducible representations of $S_n$......Page 281
5.5 Characters. Orthogonality relations......Page 285
5.6 Direct products of groups. Characters of abelian groups......Page 295
5.7 Some arithmetical considerations......Page 298
5.8 Burnside's $p^a q^b$ theorem......Page 300
5.9 Induced modules......Page 302
5.10 Properties of induction. Frobenius reciprocity theorem......Page 308
5.11 Further results on induced modules......Page 315
5.12 Brauer's theorem on induced characters......Page 321
5.13 Brauer's theorem on splitting fields......Page 329
5.14 The Schur index......Page 330
5.15 Frobenius groups......Page 333
References......Page 341
6 ELEMENTS OF HOMOLOGICAL ALGEBRA WITH APPLICATIONS......Page 342
6.1 Additive and abelian categories......Page 343
6.2 Complexes and homology......Page 347
6.3 Long exact homology sequence......Page 350
6.4 Homotopy......Page 353
6.5 Resolutions......Page 355
6.6 Derived functors......Page 358
6.7 Ext......Page 362
6.8 Tor......Page 369
6.9 Cohomology of groups......Page 371
6.10 Extensions of groups......Page 379
6.11 Cohomology of algebras......Page 386
6.12 Homological dimension......Page 391
6.13 Koszul's complex and Hilbert's syzygy theorem......Page 394
References......Page 403
7 COMMUTATIVE IDEAL THEORY: GENERAL THEORY AND NOETHERIAN RINGS......Page 404
7.1 Prime ideals. Nil radical......Page 405
7.2 Localization of rings......Page 409
7.3 Localization of modules......Page 413
7.4 Localization at the complement of a prime ideal. Local-global relations......Page 416
7.5 Prime spectrum of a commutative ring......Page 419
7.6 Integral dependence......Page 424
7.7 Integrally closed domains......Page 428
7.8 Rank of projective modules......Page 430
7.9 Projective class group......Page 435
7.10 Noetherian rings......Page 436
7.11 Commutative artinian rings......Page 441
7.12 Affine algebraic varieties. The Hilbert Nullstellensatz......Page 443
7.13 Primary decompositions......Page 449
7.14 Artin-Rees lemma. Krull intersection theorem......Page 456
7.15 Hilbert's polynomial for a graded module......Page 459
7.16 The characteristic polynomial of a noetherian local ring......Page 464
7.17 Krull dimension......Page 466
7.18 $I$-adic topologies and completions......Page 471
References......Page 478
8 FIELD THEORY......Page 479
8.1 Algebraic closure of a field......Page 480
8.2 The Jacobson-Bourbaki correspondence......Page 484
8.3 Finite Galois theory......Page 487
8.4 Crossed products and the Brauer group......Page 491
8.5 Cyclic algebras......Page 500
8.6 Infinite Galois theory......Page 502
8.7 Separability and normality......Page 505
8.8 Separable splitting fields......Page 511
8.9 Kummer extensions......Page 514
8.10 Rings of Witt vectors......Page 517
8.11 Abelian $p$-extension......Page 525
8.12 Transcendency bases......Page 530
8.13 Transcendency bases for domains. Affine algebras......Page 533
8.14 Luroth's theorem......Page 536
8.15 Separability for arbitrary extension fields......Page 541
8.16 Derivations......Page 546
8.17 Galois theory for purely inseparable extensions of exponent one......Page 557
8.18 Tensor products of fields......Page 560
8.19 Free composites of fields......Page 566
References......Page 572
9 VALUATION THEORY......Page 573
9.1 Absolute values......Page 574
9.2 The approximation theorem......Page 578
9.3 Absolute values on $\mathbb{Q}$ and $F(x)$......Page 580
9.4 Completion of a field......Page 582
9.5 Finite dimensional extensions of complete fields. The archimedean case......Page 585
9.6 Valuations......Page 589
9.7 Valuation rings and places......Page 593
9.8 Extension of homomorphisms and valuations......Page 596
9.9 Determination of the absolute values of a finite dimensional extension field......Page 601
9.10 Ramification index and residue degree. Discrete valuations......Page 604
9.11 Hensel's lemma......Page 608
9.12 Local fields......Page 611
9.13 Totally disconnected locally compact division rings......Page 615
9.14 The Brauer group of a local field......Page 624
9.15 Quadratic forms over local fields......Page 627
References......Page 634
10 DEDEKIND DOMAINS......Page 635
10.1 Fractional ideals. Dedekind domains......Page 636
10.2 Characterizations of Dedekind domains......Page 641
10.3 Integral extensions of Dedekind domains......Page 647
10.4 Connections with valuation theory......Page 650
10.5 Ramified primes and the discriminant......Page 655
10.6 Finitely generated modules over a Dedekind domain......Page 659
References......Page 665
11 FORMALLY REAL FIELDS......Page 666
11.1 Formally real fields......Page 667
11.2 Real closures......Page 671
11.3 Totally positive elements......Page 673
11.4 Hilbert's seventeenth problem......Page 676
11.5 Pfister theory of quadratic forms......Page 679
11.6 Sums of squares in $R(x_1, \ldots ,x_n)$, $R$ a real closed field......Page 685
11.7 Artin-Schreier characterization of real closed fields......Page 690
References......Page 693
INDEX......Page 695