AlgebraThe intent of this book is to introduce readers to algebra from a point of view that stresses examples and classification. Whenever possible, the main theorems are treated as tools that may be used to construct and analyze specific types of groups, rings, fields, modules, etc. Sample constructions and classifications are given in both text and exercises. |
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A-module homomorphism a₁ abelian group algebraic closure Ann(M automorphism b₁ basis canonical map coefficients commutative ring conjugate Corollary cyclic Dedekind domain defined Definition degree direct sum divides division ring element exact sequence Exercises exponent finite extension finite group finitely generated projective free modules function functor G-set Gal(K/k Galois extension Galois group gives group G group homomorphism group of order hence homomorphism f identity inclusion induced injective integers integral domain inverse irreducible K₁ kernel left A-module left ideal Lemma Let f Let G m₁ matrix maximal ideal minimal polynomial Mn(A monoid morphisms multiplication Noetherian nontrivial nonzero normal prime ideal primitive n-th root projective module Proof Let Proposition ring and let ring homomorphism root of unity solvable splitting field subgroup of G submodule subset suffices to show Suppose given surjective Theorem two-sided ideal unique universal property vector space write