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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 | |