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| Preface | |
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| Preface to the First Edition | |
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| Introduction: Concepts from set Theory. The Integers | |
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| The power set of a set | |
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| The Cartesian product set. Maps | |
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| Equivalence relations. Factoring a map through an equivalence relation | |
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| The natural numbers | |
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| The number system Z of integers | |
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| Some basic arithmetic facts about Z | |
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| A word on cardinal numbers | |
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| Monoids and Groups | |
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| Monoids of transformations and abstract monoids | |
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| Groups of transformations and abstract groups | |
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| Isomorphism. Cayley's theorem | |
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| Generalized associativity. Commutativity | |
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| Submonoids and subgroups generated by a subset. Cyclic groups | |
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| Cycle decomposition of permutations | |
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| Orbits. Cosets of a subgroup | |
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| Congruences. Quotient monoids and groups | |
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| Homomorphisms | |
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| Subgroups of a homomorphic image. Two basic isomorphism theorems | |
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| Free objects. Generators and relations | |
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| Groups acting on sets | |
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| Sylow's theorems | |
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| Rings | |
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| Definition and elementary properties | |
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| Types of rings | |
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| Matrix rings | |
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| Quaternions | |
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| Ideals, quotient rings | |
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| Ideals and quotient rings for Z | |
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| Homomorphisms of rings. Basic theorems | |
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| Anti-isomorphisms | |
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| Field of fractions of a commutative domain | |
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| Polynomial rings | |
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| Some properties of polynomial rings and applications | |
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| Polynomial functions | |
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| Symmetric polynomials | |
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| Factorial monoids and rings | |
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| Principal ideal domains and Euclidean domains | |
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| Polynomial extensions of factorial domains | |
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| "Rngs" (rings without unit) | |
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| Modules over a Principal Ideal Domain | |
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| Ring of endomorphisms of an abelian group | |
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| Left and right modules | |
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| Fundamental concepts and results | |
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| Free modules and matrices | |
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| Direct sums of modules | |
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| Finitely generated modules over a p.i.d. Preliminary results | |
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| Equivalence of matrices with entries in a p.i.d | |
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| Structure theorem for finitely generated modules over a p.i.d | |
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| Torsion modules, primary components, invariance theorem | |
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| Applications to abelian groups and to linear transformations | |
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| The ring of endomorphisms of a finitely generated module over a p.i.d | |
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| Galois Theory of Equations | |
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| Preliminary results, some old, some new | |
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| Construction with straight-edge and compass | |
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| Splitting field of a polynomial | |
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| Multiple roots | |
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| The Galois group. The fundamental Galois pairing | |
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| Some results on finite groups | |
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| Galois' criterion for solvability by radicals | |
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| The Galois group as permutation group of the roots | |
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| The general equation of the nth degree | |
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| Equations with rational coefficients and symmetric group as Galois group | |
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| Constructible regular n-gons | |
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| Transcendence of e and p. The Lindemann-Weierstrass theorem | |
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| Finite fields | |
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| Special bases for finite dimensional extensions fields | |
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| Traces and norms | |
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| Mod p reduction | |
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| Real Polynomial Equations and Inequalities | |
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| Ordered fields. Real closed fields | |
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| Sturm's theorem | |
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| Formalized Euclidean algorithm and Sturm's theorem | |
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| Elimination procedures. Resultants | |
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| Decision method for an algebraic curve | |
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| Tarski's theorem | |
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| Metric Vector Spaces and the Classical Groups | |
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| Linear functions and bilinear forms | |
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| Alternate forms | |
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| Quadratic forms and symmetric bilinear forms | |
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| Basic concepts of orthogonal geometry | |
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| Witt's cancellation theorem | |
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| The theorem of Cartan-Dieudonne | |
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| Structure of the general linear group GLn(F) | |
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| Structure of orthogonal groups | |
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| Symplectic geometry. The symplectic group | |
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| Orders of orthogonal and symplectic groups over a finite field | |
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| Postscript on hermitian forms and unitary geometry | |
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| Algebras over a Field | |
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| Definition and examples of associative algebras | |
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| Exterior algebras. Application to determinants | |
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| Regular matrix representations of associative algebras. Norms and traces | |
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| Change of base field. Transitivity of trace and norm | |
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| Non-associative algebras. Lie and Jordan algebras | |
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| Hurwitz' problem. Composition algebras | |
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| Frobenius' and Wedderburn's theorems on associative division algebras | |
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| Lattices and Boolean Algebras | |
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| Partially ordered sets and lattices | |
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| Distributivity and modularity | |
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| The theorem of Jordan-Holder-Dedekind | |
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| The lattice of subspaces of a vector space. Fundamental theorem of projective geometry | |
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| Boolean algebras | |
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| The Mobius function of a partially ordered set | |
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| Appendix | |
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| Index | |