## A Guide to Groups, Rings, and FieldsAlgebraic structures have come to be ubiquitous in mathematics, with almost all mathematicians encountering groups, rings, fields or more exotic related objects during the course of their research. This book presents an overview of some of the most important algebraic structures in modern mathematics, with an emphasis on creating a coherent picture of how they all interact. In addition to the standard material on groups, rings, modules, fields, and Galois theory, the book includes discussions of other important topics, including linear groups, group representations, Artinian rings, projective, injective and flat modules, Dedekind domains, and central simple algebras. All of the important theorems are discussed, typically without proofs, but often with a discussion of the intuitive ideas behind those proofs. This insightful guide is ideal for both graduate students in mathematics who are beginning their studies, and researchers who wish to understand the bigger picture of the algebraic structures they encounter. |

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abelian groups action algebraic closure arrows Artinian automorphism basis called central simple algebra commutative ring conjugacy classes contains cosets cyclic define Definition degree denote direct sum division ring elements exact sequence example field extension finite group free modules function functor Gal.F=K Galois extension Galois group given GL.n group G homomorphism f identity ifand ifit infinite injective integer inverse irreducible isomorphic K-algebra kernel KŒG KŒX KŒX1 X2 left ideal left R-module let f Let G Let K F linear Mathematical matrix monoid morphism multiplication nilpotent Noetherian noncommutative nonzero normal subgroup object permutation phism prime ideal projective quotient R-linear R-module representation right R-module ring and let ring homomorphism ring of polynomials roots semisimple skew field solvable splitting field subfield submodule subring subset Suppose surjective tensor product Theorem topology torsion two-sided ideal unique variables vector space zero