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Basic algebra / Nathan Jacobson

Jacobson, Nathan, 1910-.
Material type: materialTypeLabelBook; Format: print Publisher: Mineola : Dover, 2009Edition: 2nd ed.Description: 2 v. ; 24 cm.ISBN: 978-0-486-47189-1 (I); 978-0-486-47187-7 (II).Subject(s): Algebra
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Manuales (7 días) 02. BIBLIOTECA CAMPUS PUERTO REAL
512/JAC/basun (Browse shelf) Vol. 1 Available   Shelving location | Bibliomaps® BIBLIOG. RECOM. 3743116341

TEORÍA DE GALOIS GRADO EN MATEMÁTICAS Asignatura actualizada 2017-2018

Manuales (7 días) 02. BIBLIOTECA CAMPUS PUERTO REAL
512/JAC/basun (Browse shelf) Vol. 1 Available   Shelving location | Bibliomaps® BIBLIOG. RECOM. 374311628X
Manuales (7 días) 02. BIBLIOTECA CAMPUS PUERTO REAL
512/JAC/basun (Browse shelf) Vol. 1 Available   Shelving location | Bibliomaps® BIBLIOG. RECOM. 3743116225
Manuales (7 días) 02. BIBLIOTECA CAMPUS PUERTO REAL
512/JAC/basun (Browse shelf) Vol. 2 Available   Shelving location | Bibliomaps® BIBLIOG. RECOM. 3743116403
Manuales (7 días) 02. BIBLIOTECA CAMPUS PUERTO REAL
512/JAC/basun (Browse shelf) Vol. 2 Available   Shelving location | Bibliomaps® BIBLIOG. RECOM. 3743116468
Manuales (7 días) 02. BIBLIOTECA CAMPUS PUERTO REAL
512/JAC/basun (Browse shelf) Vol. 2 Available   Shelving location | Bibliomaps® BIBLIOG. RECOM. 374311652X
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Enhanced descriptions from Syndetics:

Volume I of a pair of classic texts -- and standard references for a generation -- this book is the work of an expert algebraist who taught at Yale for two decades. Volume I covers all undergraduate topics, including groups, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra. 1985 edition.

Índice

Nathan Jacobson's Basic Algebra I does not refer to basic, beginner algebra, but rather the first course one takes after linear algebra. This dense text provides both instruction and practice in understanding concepts from set theory, monoids and groups, rings, modules over a principal ideal domain, Galois Theory of equations, real polynomial equations and inequalities, metric vector spaces and the classical groups, algebras over a field, and lattices and Boolean algebras. Carefully explained proofs are also included.

Table of contents provided by Syndetics

  • Preface (p. xi)
  • Preface to the First Edition (p. xiii)
  • Introduction: Concepts from set Theory. The Integers (p. 1)
  • 0.1 The power set of a set (p. 3)
  • 0.2 The Cartesian product set. Maps (p. 4)
  • 0.3 Equivalence relations. Factoring a map through an equivalence relation (p. 10)
  • 0.4 The natural numbers (p. 15)
  • 0.5 The number system Z of integers (p. 19)
  • 0.6 Some basic arithmetic facts about Z (p. 22)
  • 0.7 A word on cardinal numbers (p. 24)
  • 1 Monoids and Groups (p. 26)
  • 1.1 Monoids of transformations and abstract monoids (p. 28)
  • 1.2 Groups of transformations and abstract groups (p. 31)
  • 1.3 Isomorphism. Cayley's theorem (p. 36)
  • 1.4 Generalized associativity. Commutativity (p. 39)
  • 1.5 Submonoids and subgroups generated by a subset. Cyclic groups (p. 42)
  • 1.6 Cycle decomposition of permutations (p. 48)
  • 1.7 Orbits. Cosets of a subgroup (p. 51)
  • 1.8 Congruences. Quotient monoids and groups (p. 54)
  • 1.9 Homomorphisms (p. 58)
  • 1.10 Subgroups of a homomorphic image. Two basic isomorphism theorems (p. 64)
  • 1.11 Free objects. Generators and relations (p. 67)
  • 1.12 Groups acting on sets (p. 71)
  • 1.13 Sylow's theorems (p. 79)
  • 2 Rings (p. 85)
  • 2.1 Definition and elementary properties (p. 86)
  • 2.2 Types of rings (p. 90)
  • 2.3 Matrix rings (p. 92)
  • 2.4 Quaternions (p. 98)
  • 2.5 Ideals, quotient rings (p. 101)
  • 2.6 Ideals and quotient rings for Z (p. 103)
  • 2.7 Homomorphisms of rings. Basic theorems (p. 106)
  • 2.8 Anti-isomorphisms (p. 111)
  • 2.9 Field of fractions of a commutative domain (p. 115)
  • 2.10 Polynomial rings (p. 119)
  • 2.11 Some properties of polynomial rings and applications (p. 127)
  • 2.12 Polynomial functions (p. 134)
  • 2.13 Symmetric polynomials (p. 138)
  • 2.14 Factorial monoids and rings (p. 140)
  • 2.15 Principal ideal domains and Euclidean domains (p. 147)
  • 2.16 Polynomial extensions of factorial domains (p. 151)
  • 2.17 "Rngs" (rings without unit) (p. 155)
  • 3 Modules over a Principal Ideal Domain (p. 157)
  • 3.1 Ring of endomorphisms of an abelian group (p. 158)
  • 3.2 Left and right modules (p. 163)
  • 3.3 Fundamental concepts and results (p. 166)
  • 3.4 Free modules and matrices (p. 170)
  • 3.5 Direct sums of modules (p. 175)
  • 3.6 Finitely generated modules over a p.i.d. Preliminary results (p. 179)
  • 3.7 Equivalence of matrices with entries in a p.i.d (p. 181)
  • 3.8 Structure theorem for finitely generated modules over a p.i.d (p. 187)
  • 3.9 Torsion modules, primary components, invariance theorem (p. 189)
  • 3.10 Applications to abelian groups and to linear transformations (p. 194)
  • 3.11 The ring of endomorphisms of a finitely generated module over a p.i.d (p. 204)
  • 4 Galois Theory of Equations (p. 210)
  • 4.1 Preliminary results, some old, some new (p. 213)
  • 4.2 Construction with straight-edge and compass (p. 216)
  • 4.3 Splitting field of a polynomial (p. 224)
  • 4.4 Multiple roots (p. 229)
  • 4.5 The Galois group. The fundamental Galois pairing (p. 234)
  • 4.6 Some results on finite groups (p. 244)
  • 4.7 Galois' criterion for solvability by radicals (p. 251)
  • 4.8 The Galois group as permutation group of the roots (p. 256)
  • 4.9 The general equation of the nth degree (p. 262)
  • 4.10 Equations with rational coefficients and symmetric group as Galois group (p. 267)
  • 4.11 Constructible regular n-gons (p. 271)
  • 4.12 Transcendence of e and p. The Lindemann-Weierstrass theorem (p. 277)
  • 4.13 Finite fields (p. 287)
  • 4.14 Special bases for finite dimensional extensions fields (p. 290)
  • 4.15 Traces and norms (p. 296)
  • 4.16 Mod p reduction (p. 301)
  • 5 Real Polynomial Equations and Inequalities (p. 306)
  • 5.1 Ordered fields. Real closed fields (p. 307)
  • 5.2 Sturm's theorem (p. 311)
  • 5.3 Formalized Euclidean algorithm and Sturm's theorem (p. 316)
  • 5.4 Elimination procedures. Resultants (p. 322)
  • 5.5 Decision method for an algebraic curve (p. 327)
  • 5.6 Tarski's theorem (p. 335)
  • 6 Metric Vector Spaces and the Classical Groups (p. 342)
  • 6.1 Linear functions and bilinear forms (p. 343)
  • 6.2 Alternate forms (p. 349)
  • 6.3 Quadratic forms and symmetric bilinear forms (p. 354)
  • 6.4 Basic concepts of orthogonal geometry (p. 361)
  • 6.5 Witt's cancellation theorem (p. 367)
  • 6.6 The theorem of Cartan-Dieudonne (p. 371)
  • 6.7 Structure of the general linear group GLn(F) (p. 375)
  • 6.8 Structure of orthogonal groups (p. 382)
  • 6.9 Symplectic geometry. The symplectic group (p. 391)
  • 6.10 Orders of orthogonal and symplectic groups over a finite field (p. 398)
  • 6.11 Postscript on hermitian forms and unitary geometry (p. 401)
  • 7 Algebras over a Field (p. 405)
  • 7.1 Definition and examples of associative algebras (p. 406)
  • 7.2 Exterior algebras. Application to determinants (p. 411)
  • 7.3 Regular matrix representations of associative algebras. Norms and traces (p. 422)
  • 7.4 Change of base field. Transitivity of trace and norm (p. 426)
  • 7.5 Non-associative algebras. Lie and Jordan algebras (p. 430)
  • 7.6 Hurwitz' problem. Composition algebras (p. 438)
  • 7.7 Frobenius' and Wedderburn's theorems on associative division algebras (p. 451)
  • 8 Lattices and Boolean Algebras (p. 455)
  • 8.1 Partially ordered sets and lattices (p. 456)
  • 8.2 Distributivity and modularity (p. 461)
  • 8.3 The theorem of Jordan-Holder-Dedekind (p. 466)
  • 8.4 The lattice of subspaces of a vector space. Fundamental theorem of projective geometry (p. 468)
  • 8.5 Boolean algebras (p. 474)
  • 8.6 The Mobius function of a partially ordered set (p. 480)
  • Appendix (p. 489)
  • Index (p. 493)

Author notes provided by Syndetics

One of the world's leading researchers in abstract algebra, Nathan Jacobson (1910-95) taught at several prominent universities, including the University of Chicago, Johns Hopkins, and Yale.

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