Abstract AlgebraCourier Corporation, 24 de maig 2012 - 656 pàgines This excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. These systems, which consist of sets of elements, operations, and relations among the elements, and prescriptive axioms, are abstractions and generalizations of various models which evolved from efforts to explain or discuss physical phenomena. In Chapter 1, the author discusses the essential ingredients of a mathematical system, and in the next four chapters covers the basic number systems, decompositions of integers, diophantine problems, and congruences. Chapters 6 through 9 examine groups, rings, domains, fields, polynomial rings, and quadratic domains.Chapters 10 through 13 cover modular systems, modules and vector spaces, linear transformations and matrices, and the elementary theory of matrices. The author, Professor of Mathematics at the University of Pittsburgh, includes many examples and, at the end of each chapter, a large number of problems of varying levels of difficulty. |
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Abelian group addition and multiplication algebraic automorphism Axiom binary operations canonical basis characteristic polynomial classes column commutative components concept conclude congruence consider contains Corollary correspondence cosets cyclic defined definition denoted Describe Determine diagonal divisors of zero elementary matrices elementary row operations elements of F equal equation equivalence relation Example expressible field F find finding finite first GF(p group G Hence identity element implies induction hypothesis infinite integral domain irreducible irreducible polynomial isomorphic Lemma linear transformation linearly independent mathematical matrix modulo monic natural numbers nonsingular nonzero elements normal subgroup number system obtain one-to-one mapping operation table pair positive integer Problem proof properties prove quadratic quotient rank rational integers rational number reduced row echelon relatively prime ring row echelon form semigroup Show solution splitting field subfield subgroup of G subgroup of order subring subset theory unique vector space vectors v1 zero mod zeros modulo