217825377-Basic-Algebra-i-Jacobson.pdf - aSlc Igebra I...

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.. aSlc Igebra I Second Edition NATHAN JACOBSON YALE UNIVERSITY rn W. H. FREEMAN AND COMPANY New York
Preface xi Preface to the First Edition xiii INTRODUCTION: CONCEPTS FROM SET THEORY. THE INTEGERS 1 0.1 The power set of a set 3 0.2 The Cartesian product set. Maps 4 0.3 Equivalence relations. Factoring a map through an equivalence relation 10 0.4 The natural numbers 15 0.5 The number system 7L of integers 19 0.6 Some basic arithmetic facts about 7L 22 0.7 A word on cardinal numbers 24 MONOIDS AND GROUPS 26 Contents ~'l '. 1.1 1.2 1.3 Monoids of transformations and abstract monoids 28 Groups of transformations and abstract groups 31 Isomorphism. Cayley's theorem 36
Contents GALOIS THEORY OF EQUATIONS 210 METRIC VECTOR SPACES AND THE CLASSICAL GROUPS ix 277 342 398 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 The 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 The Galois group as permutation group of the roots 256 4.9 The 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 n. The Lindemann-Weierstrass theorem 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 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 6.1 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 The 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 6.11 Postscript on hermitian forms and unitary geometry 401 ALGEBRAS OVER A FIELD 405 7.1 Definition and examples of associative algebras 406 7.2 Exterior algebras. Application to determinants 411 4 5 6 7 Contents MODULES OVER A PRINCIPAL IDEAL DOMAIN 157 RINGS 85 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 1.9 Homomorphisms 58 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 2.2 Types of rings 90 2.3 Matrix rings 92 2.4 Quaternions 98 2.5 Ideals, quotient rings 101 2.6 Ideals and quotient rings for 7l 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

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