Algebra fulfills a definite need to provide a self-contained, one volume, graduate level algebra text that is readable by the average graduate student and flexible enough to accomodate a wide variety of instructors and course contents. The guiding philosophical principle throughout the text is that the material should be presented in the maximum usable generality consistent with good pedagogy. Therefore it is essentially self-contained, stresses clarity rather than brevity and contains an unusually large number of illustrative exercises. The book covers major areas of modern algebra, which is a necessity for most mathematics students in sufficient breadth and depth.
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Prerequisites and Preliminaries
Relations and Partitions
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abelian group algebraically closed automorphism basis chain condition char commutative ring Consequently contains Corollary coset cyclic defined Definition denoted direct product direct sum disjoint division ring divisors endomorphism epimorphism equivalent EXAMPLE Exercise exists extension field finite dimensional finite number free module function functor Galois group given group G hence Hint implies infinite integral domain intermediate field invertible isomorphism left Artinian left ideal left module Lemma Let F Let G linear linearly independent matrix module homomorphism monic monomorphism morphism multiplicative nilpotent Noetherian nonempty normal subgroup phism positive integer prime ideal primitive principal ideal domain proof of Theorem Proposition prove quotient R-module radical REMARK resp ring with identity Section semisimple SKETCH OF PROOF solvable splitting field subfield subgroup of G submodule subring subset Sylow Theorem 1.6 transcendence base vector space Verify whence zero