Introduction: Concepts from set Theory. The Integers 1 --
0.1 Power set of a set 3 --
0.2 Cartesian product set. Maps 4 --
0.3 Equivalence relations. Factoring a map through an equivalence relation 10 --
0.4 Natural numbers 15 --
0.5 Number system Z of integers 19 --
0.6 Some basic arithmetic facts about Z 22 --
0.7 A word on cardinal numbers 24 --
1 Monoids and Groups 26 --
1.1 Monoids of transformations and abstract monoids 28 --
1.2 Groups of transformations and abstract groups 31 --
1.3 Isomorphism. Cayley's theorem 36 --
1.4 Generalized associativity. Commutativity 39 --
1.5 Submonoids and subgroups generated by a subset. Cyclic groups 42 --
1.6 Cycle decomposition of permutations 48 --
1.7 Orbits. Cosets of a subgroup 51 --
1.8 Congruences. Quotient monoids and groups 54 --
1.10 Subgroups of a homomorphic image. Two basic isomorphism theorems 64 --
1.11 Free objects. Generators and relations 67 --
1.12 Groups acting on sets 71 --
1.13 Sylow's theorems 79 --
2.1 Definition and elementary properties 86 --
2.5 Ideals, quotient rings 101 --
2.6 Ideals and quotient rings for Z 103 --
2.7 Homomorphisms of rings. Basic theorems 106 --
2.8 Anti-isomorphisms 111 --
2.9 Field of fractions of a commutative domain 115 --
2.10 Polynomial rings 119 --
2.11 Some properties of polynomial rings and applications 127 --
2.12 Polynomial functions 134 --
2.13 Symmetric polynomials 138 --
2.14 Factorial monoids and rings 140 --
2.15 Principal ideal domains and Euclidean domains 147 --
2.16 Polynomial extensions of factorial domains 151 --
2.17 "Rngs" (rings without unit) 155 --
3 Modules over a Principal Ideal Domain 157 --
3.1 Ring of endomorphisms of an abelian group 158 --
3.2 Left and right modules 163 --
3.3 Fundamental concepts and results 166 --
3.4 Free modules and matrices 170 --
3.5 Direct sums of modules 175 --
3.6 Finitely generated modules over a p.i.d. Preliminary results 179 --
3.7 Equivalence of matrices with entries in a p.i.d 181 --
3.8 Structure theorem for finitely generated modules over a p.i.d 187 --
3.9 Torsion modules, primary components, invariance theorem 189 --
3.10 Applications to abelian groups and to linear transformations 194 --
3.11 Ring of endomorphisms of a finitely generated module over a p.i.d 204 --
4 Galois Theory of Equations 210 --
4.1 Preliminary results, some old, some new 213 --
4.2 Construction with straight-edge and compass 216 --
4.3 Splitting field of a polynomial 224 --
4.4 Multiple roots 229 --
4.5 Galois group. The fundamental Galois pairing 234 --
4.6 Some results on finite groups 244 --
4.7 Galois' criterion for solvability by radicals 251 --
4.8 Galois group as permutation group of the roots 256 --
4.9 General equation of the nth degree 262 --
4.10 Equations with rational coefficients and symmetric group as Galois group 267 --
4.11 Constructible regular n-gons 271 --
4.12 Transcendence of e and p. The Lindemann-Weierstrass theorem 277 --
4.13 Finite fields 287 --
4.14 Special bases for finite dimensional extensions fields 290 --
4.15 Traces and norms 296 --
4.16 Mod p reduction 301 --
5 Real Polynomial Equations and Inequalities 306 --
5.1 Ordered fields. Real closed fields 307 --
5.2 Sturm's theorem 311 --
5.3 Formalized Euclidean algorithm and Sturm's theorem 316 --
5.4 Elimination procedures. Resultants 322 --
5.5 Decision method for an algebraic curve 327 --
5.6 Tarski's theorem 335 --
6 Metric Vector Spaces and the Classical Groups 342 --
6.1 Linear functions and bilinear forms 343 --
6.2 Alternate forms 349 --
6.3 Quadratic forms and symmetric bilinear forms 354 --
6.4 Basic concepts of orthogonal geometry 361 --
6.5 Witt's cancellation theorem 367 --
6.6 Theorem of Cartan-Dieudonne 371 --
6.7 Structure of the general linear group GLn(F) 375 --
6.8 Structure of orthogonal groups 382 --
6.9 Symplectic geometry. The symplectic group 391 --
6.10 Orders of orthogonal and symplectic groups over a finite field 398 --
6.11 Postscript on hermitian forms and unitary geometry 401 --
7 Algebras over a Field 405 --
7.1 Definition and examples of associative algebras 406 --
7.2 Exterior algebras. Application to determinants 411 --
7.3 Regular matrix representations of associative algebras. Norms and traces 422 --
7.4 Change of base field. Transitivity of trace and norm 426 --
7.5 Non-associative algebras. Lie and Jordan algebras 430 --
7.6 Hurwitz' problem. Composition algebras 438 --
7.7 Frobenius' and Wedderburn's theorems on associative division algebras 451 --
8 Lattices and Boolean Algebras 455 --
8.1 Partially ordered sets and lattices 456 --
8.2 Distributivity and modularity 461 --
8.3 Theorem of Jordan-Holder-Dedekind 466 --
8.4 Lattice of subspaces of a vector space. Fundamental theorem of projective geometry 468 --
8.5 Boolean algebras 474 --
8.6 Mobius function of a partially ordered set 480.
