| Preface | p. xi |
| Preface to the First Edition | p. xiii |
| Introduction: Concepts from set Theory. The Integers | p. 1 |
| The power set of a set | p. 3 |
| The Cartesian product set. Maps | p. 4 |
| Equivalence relations. Factoring a map through an equivalence relation | p. 10 |
| The natural numbers | p. 15 |
| The number system Z of integers | p. 19 |
| Some basic arithmetic facts about Z | p. 22 |
| A word on cardinal numbers | p. 24 |
| Monoids and Groups | p. 26 |
| Monoids of transformations and abstract monoids | p. 28 |
| Groups of transformations and abstract groups | p. 31 |
| Isomorphism. Cayley's theorem | p. 36 |
| Generalized associativity. Commutativity | p. 39 |
| Submonoids and subgroups generated by a subset. Cyclic groups | p. 42 |
| Cycle decomposition of permutations | p. 48 |
| Orbits. Cosets of a subgroup | p. 51 |
| Congruences. Quotient monoids and groups | p. 54 |
| Homomorphisms | p. 58 |
| Subgroups of a homomorphic image. Two basic isomorphism theorems | p. 64 |
| Free objects. Generators and relations | p. 67 |
| Groups acting on sets | p. 71 |
| Sylow's theorems | p. 79 |
| Rings | p. 85 |
| Definition and elementary properties | p. 86 |
| Types of rings | p. 90 |
| Matrix rings | p. 92 |
| Quaternions | p. 98 |
| Ideals, quotient rings | p. 101 |
| Ideals and quotient rings for Z | p. 103 |
| Homomorphisms of rings. Basic theorems | p. 106 |
| Anti-isomorphisms | p. 111 |
| Field of fractions of a commutative domain | p. 115 |
| Polynomial rings | p. 119 |
| Some properties of polynomial rings and applications | p. 127 |
| Polynomial functions | p. 134 |
| Symmetric polynomials | p. 138 |
| Factorial monoids and rings | p. 140 |
| Principal ideal domains and Euclidean domains | p. 147 |
| Polynomial extensions of factorial domains | p. 151 |
| "Rngs" (rings without unit) | p. 155 |
| Modules over a Principal Ideal Domain | p. 157 |
| Ring of endomorphisms of an abelian group | p. 158 |
| Left and right modules | p. 163 |
| Fundamental concepts and results | p. 166 |
| Free modules and matrices | p. 170 |
| Direct sums of modules | p. 175 |
| Finitely generated modules over a p.i.d. Preliminary results | p. 179 |
| Equivalence of matrices with entries in a p.i.d | p. 181 |
| Structure theorem for finitely generated modules over a p.i.d | p. 187 |
| Torsion modules, primary components, invariance theorem | p. 189 |
| Applications to abelian groups and to linear transformations | p. 194 |
| The ring of endomorphisms of a finitely generated module over a p.i.d | p. 204 |
| Galois Theory of Equations | p. 210 |
| Preliminary results, some old, some new | p. 213 |
| Construction with straight-edge and compass | p. 216 |
| Splitting field of a polynomial | p. 224 |
| Multiple roots | p. 229 |
| The Galois group. The fundamental Galois pairing | p. 234 |
| Some results on finite groups | p. 244 |
| Galois' criterion for solvability by radicals | p. 251 |
| The Galois group as permutation group of the roots | p. 256 |
| The general equation of the nth degree | p. 262 |
| Equations with rational coefficients and symmetric group as Galois group | p. 267 |
| Constructible regular n-gons | p. 271 |
| Transcendence of e and p. The Lindemann-Weierstrass theorem | p. 277 |
| Finite fields | p. 287 |
| Special bases for finite dimensional extensions fields | p. 290 |
| Traces and norms | p. 296 |
| Mod p reduction | p. 301 |
| Real Polynomial Equations and Inequalities | p. 306 |
| Ordered fields. Real closed fields | p. 307 |
| Sturm's theorem | p. 311 |
| Formalized Euclidean algorithm and Sturm's theorem | p. 316 |
| Elimination procedures. Resultants | p. 322 |
| Decision method for an algebraic curve | p. 327 |
| Tarski's theorem | p. 335 |
| Metric Vector Spaces and the Classical Groups | p. 342 |
| Linear functions and bilinear forms | p. 343 |
| Alternate forms | p. 349 |
| Quadratic forms and symmetric bilinear forms | p. 354 |
| Basic concepts of orthogonal geometry | p. 361 |
| Witt's cancellation theorem | p. 367 |
| The theorem of Cartan-Dieudonne | p. 371 |
| Structure of the general linear group GLn(F) | p. 375 |
| Structure of orthogonal groups | p. 382 |
| Symplectic geometry. The symplectic group | p. 391 |
| Orders of orthogonal and symplectic groups over a finite field | p. 398 |
| Postscript on hermitian forms and unitary geometry | p. 401 |
| Algebras over a Field | p. 405 |
| Definition and examples of associative algebras | p. 406 |
| Exterior algebras. Application to determinants | p. 411 |
| Regular matrix representations of associative algebras. Norms and traces | p. 422 |
| Change of base field. Transitivity of trace and norm | p. 426 |
| Non-associative algebras. Lie and Jordan algebras | p. 430 |
| Hurwitz' problem. Composition algebras | p. 438 |
| Frobenius' and Wedderburn's theorems on associative division algebras | p. 451 |
| Lattices and Boolean Algebras | p. 455 |
| Partially ordered sets and lattices | p. 456 |
| Distributivity and modularity | p. 461 |
| The theorem of Jordan-Holder-Dedekind | p. 466 |
| The lattice of subspaces of a vector space. Fundamental theorem of projective geometry | p. 468 |
| Boolean algebras | p. 474 |
| The Mobius function of a partially ordered set | p. 480 |
| Appendix | p. 489 |
| Index | p. 493 |
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