Basic algebra 1

Title page......Page 1
Date-line......Page 2
Dedication......Page 3
Contents......Page 5
Preface......Page 9
Preface to the First Edition......Page 11
0 INTRODUCTION: CONCEPTS FROM SET THEORY. THE INTEGERS......Page 19
0.1 The power set of a set......Page 21
0.2 The Cartesian product set Maps......Page 22
0.3 Equivalence relations. Factoring a map through an equivalence relation......Page 28
0.4 The natural numbers......Page 33
0.5 The number system $\mathbb{Z}$ of integers......Page 37
0.6 Some basic arithmetic facts about $\mathbb{Z}$......Page 40
0.7 A word on cardinal numbers......Page 42
1 MONOIDS AND GROUPS......Page 44
1.1 Monoids of transformations and abstract monoids......Page 46
1.2 Groups of transformations and abstract groups......Page 49
1.3 Isomorphism. Cayley's theorem......Page 54
1.4 Generalized associativity. Commutativity......Page 57
1.5 Submonoids and subgroups generated by a subset. Cyclic groups......Page 60
1.6 Cycle decomposition of permutations......Page 66
1.7 Orbits. Cosets of a subgroup......Page 69
1.8 Congruences. Quotient monoids and groups......Page 72
1.9 Homomorphisms......Page 76
1.10 Subgroups of a homomorphic image. Two basic isomorphism theorems......Page 82
1.11 Free objects. Generators and relations......Page 85
1.12 Groups acting on sets......Page 89
1.13 Sylow's theorems......Page 97
2 RINGS......Page 103
2.1 Definition and elementary properties......Page 104
2.2 Types of rings......Page 108
2.3 Matrix rings......Page 110
2.4 Quaternions......Page 116
2.5 Ideals, quotient rings......Page 119
2.6 Ideals and quotient rings for $\mathbb{Z}$......Page 121
2.7 Homomorphisms of rings. Basic theorems......Page 124
2.8 Anti-isomorphisms......Page 129
2.9 Field of fractions of a commutative domain......Page 133
2.10 Polynomial rings......Page 137
2.11 Some properties of polynomial rings and applications......Page 145
2.12 Polynomial functions......Page 152
2.13 Symmetric polynomials......Page 156
2.14 Factorial monoids and rings......Page 158
2.15 Principal ideal domains and Euclidean domains......Page 165
2.16 Polynomial extensions of factorial domains......Page 169
2.17 "Rngs" (rings without unit)......Page 173
3 MODULES OVER A PRINCIPAL IDEAL DOMAIN......Page 175
3.1 Ring of endomorphisms of an abelian group......Page 176
3.2 Left and right modules......Page 181
3.3 Fundamental concepts and results......Page 184
3.4 Free modules and matrices......Page 188
3.5 Direct sums of modules......Page 193
3.6 Finitely generated modules over a p.i.d. Preliminary results......Page 197
3.7 Equivalence of matrices with entries in a p.i.d......Page 199
3.8 Structure theorem for finitely generated modules over a p.i.d......Page 205
3.9 Torsion modules, primary components, invariance theorem......Page 207
3.10 Applications to abelian groups and to linear transformations......Page 212
3.11 The ring of endomorphisms of a finitely generated module over a p.i.d......Page 222
4 GALOIS THEORY OF EQUATIONS......Page 228
4.1 Preliminary results, some old, some new......Page 231
4.2 Construction with straight-edge and compass......Page 234
4.3 Splitting field of a polynomial......Page 242
4.4 Multiple roots......Page 247
4.5 The Galois group. The fundamental Galois pairing......Page 252
4.6 Some results on finite groups......Page 262
4.7 Galois' criterion for solvability by radicals......Page 269
4.8 The Galois group as permutation group of the roots......Page 274
4.9 The general equation of the nth degree......Page 280
4.10 Equations with rational coefficients and symmetric group as Galois group......Page 285
4.11 Constructible regular $n$-gons......Page 289
4.12 Transcendence of $e$ and $\pi$. The Lindemann-Weierstrass theorem......Page 295
4.13 Finite fields......Page 305
4.14 Special bases for finite dimensional extensions fields......Page 308
4.15 Traces and norms......Page 314
4.16 Mod $p$ reduction......Page 319
5 REAL POLYNOMIAL EQUATIONS AND INEQUALITIES......Page 324
5.1 Ordered fields. Real closed fields......Page 325
5.2 Sturm's theorem......Page 329
5.3 Formalized Euclidean algorithm and Sturm's theorem......Page 334
5.4 Elimination procedures Resultants......Page 340
5.5 Decision method for an algebraic curve......Page 345
5.6 Tarski's theorem......Page 353
6 METRIC VECTOR SPACES AND THE CLASSICAL GROUPS......Page 360
6.1 Linear functions and bilinear forms......Page 361
6.2 Alternate forms......Page 367
6.3 Quadratic forms and symmetric bilinear forms......Page 372
6.4 Basic concepts of orthogonal geometry......Page 379
6.5 Witt's cancellation theorem......Page 385
6.6 The theorem of Cartan-Dieudonne......Page 389
6.7 Structure of the general linear group $GL_n(F)$......Page 393
6.8 Structure of orthogonal groups......Page 400
6.9 Symplectic geometry. The symplectic group......Page 409
6.10 Orders of orthogonal and symplectic groups over a finite field......Page 416
6.11 Postscript on hermitian forms and unitary geometry......Page 419
7 ALGEBRAS OVER A FIELD......Page 423
7.1 Definition and examples of associative algebras......Page 424
7.2 Exterior algebras. Application to determinants......Page 429
7.3 Regular matrix representations of associative algebras. Norms and traces......Page 440
7.4 Change of base field. Transitivity of trace and norm......Page 444
7.5 Non-associative algebras. Lie and Jordan algebras......Page 448
7.6 Hurwitz' problem. Composition algebras......Page 456
7.7 Frobenius' and Wedderburn's theorems on associative division algebras......Page 469
8 LATTICES AND BOOLEAN ALGEBRAS......Page 473
8.1 Partially ordered sets and lattices......Page 474
8.2 Distributivity and modularity......Page 479
8.3 The theorem of Jordan-Hoelder-Dedekind......Page 484
8.4 The lattice of subspaces of a vector space. Fundamental theorem of projective geometry......Page 486
8.5 Boolean algebras......Page 492
8.6 The Moebius function of a partially ordered set......Page 498
Appendix......Page 507
Index......Page 511