Basic algebra I

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Converted from the djvu edition, and added bookmarks. Title Contents Preface Preface to the First Edition INTRODUCTION: CONCEPTS FROM SET THEORY. THE INTEGERS 0.1 The power set of a set 0.2 The Cartesian product set. Maps 0.3 Equivalence relations. Factoring a map through an equivalence relation 0.4 The natural numbers 0.5 The number system $\mathbb{Z}$ of integers 0.6 Some basic arithmetic facts about $\mathbb{Z}$ 0.7 A word on cardinal numbers 1 MONOIDS AND GROUPS 1.1 Monoids of transformations and abstract monoids 1.2 Groups of transformations and abstract groups 1.3 Isomorphism. Cayley's theorem 1.4 Generalized associativity. Commutativity 1.5 Submonoids and subgroups generated by a subset. Cyclic groups 1.6 Cycle decomposition of permutations 1.7 Orbits. Cosets of a subgroup 1.8 Congruences. Quotient monoids and groups 1.9 Homomorphisms 1.10 Subgroups of a homomorphic image. Two basic isomorphism theorems 1.11 Free objects. Generators and relations 1.12 Groups acting on sets 1.13 Sylow's theorems 2 RINGS 2.1 Definition and elementary properties 2.2 Types of rings 2.3 Matrix rings 2.4 Quaternions 2.5 Ideals, quotient rings 2.6 Ideals and quotient rings for $\mathbb{Z}$ 2.7 Homomorphisms of rings. Basic theorems 2.8 Anti-isomorphisms 2.9 Field of fractions of a commutative domain 2.10 Polynomial rings 2.11 Some properties of polynomial rings and applications 2.12 Polynomial functions 2.13 Symmetric polynomials 2.14 Factorial monoids and rings 2.15 Principal ideal domains and Euclidean domains 2.16 Polynomial extensions of factorial domains 2.17 "Rngs" (rings without unit) 3 MODULES OVER A PRINCIPAL IDEAL DOMAIN 3.1 Ring of endomorphisms of an abelian group 3.2 Left and right modules 3.3 Fundamental concepts and results 3.4 Free modules and matrices 3.5 Direct sums of modules 3.6 Finitely generated modules over a p.i.d. Preliminary results 3.7 Equivalence of matrices with entries in a p.i.d. 3.8 Structure theorem for finitely generated modules over a p.i.d. 3.9 Torsion modules, primary components, invariance theorem 3.10 Applications to abelian groups and to linear transformations 3.11 The ring of endomorphisms of a finitely generated module over a p.i.d. 4 GALOIS THEORY OF EQUATIONS 4.1 Preliminary results, some old, some new 4.2 Construction with straight-edge and compass 4.3 Splitting field of a polynomial 4.4 Multiple roots 4.5 The Galois group. The fundamental Galois pairing 4.6 Some results on finite groups 4.7 Galois' criterion for solvability by radicals 4.8 The Galois group as permutation group of the roots 4.9 The general equation of the nth degree 4.10 Equations with rational coefficients and symmetric group as Galois group 4.11 Constructible regular n-gons 4.12 Transcendence of e and π. The Lindemann-Weierstrass theorem 4.13 Finite fields 4.14 Special bases for finite dimensional extensions fields 4.15 Traces and norms 4.16 Mod p reduction 5 REAL POLYNOMIAL EQUATIONS AND INEQUALITIES 5.1 Ordered fields. Real closed fields 5.2 Sturm's theorem 5.3 Formalized Euclidean algorithm and Sturm's theorem 5.4 Elimination procedures. Resultants 5.5 Decision method for an algebraic curve 5.6 Tarski's theorem 6 METRIC VECTOR SPACES AND THE CLASSICAL GROUPS 6.1 Linear functions and bilinear forms 6.2 Alternate forms 6.3 Quadratic forms and symmetric bilinear forms 6.4 Basic concepts of orthogonal geometry 6.5 Witt's cancellation theorem 6.6 The theorem of Cartan-Dieudonné 6.7 Structure of the general linear group $GL_n(F)$ 6.8 Structure of orthogonal groups 6.9 Symplectic geometry. The symplectic group 6.10 Orders of orthogonal and symplectic groups over a finite field 6.11 Postscript on hermitian forms and unitary geometry 7 ALGEBRAS OVER A FIELD 7.1 Definition and examples of associative algebras 7.2 Exterior algebras. Application to determinants 7.3 Regular matrix representations of associative algebras. Norms and traces 7.4 Change of base field. Transitivity of trace and norm 7.5 Non-associative algebras. Lie and Jordan algebras 7.6 Hurwitz' problem. Composition algebras 7.7 Frobenius' and Wedderburn's theorems on associative division algebras 8 LATTICES AND BOOLEAN ALGEBRAS 8.1 Partially ordered sets and lattices 8.2 Distributivity and modularity 8.3 The theorem of Jordan-Hölder-Dedekind 8.4 The lattice of subspaces of a vector space. Fundamental theorem of projective geometry 8.5 Boolean algebras 8.6 The Mobius function of a partially ordered set Appendix Index