Tóm tắt

Preliminaries -- Group Theory. Introduction to groups -- Subgroups -- Quotient groups and homomorphisms -- Group actions -- Direct and semidirect products and abelian groups -- Further topics in group theory -- Ring Theory. Introduction to rings -- Euclidean domains, principal ideal domains and unique factorization domains -- Polynomial rings -- Modules and Vector Spaces. Introduction to module theory -- Vector spaces -- Modules over principal ideal domains -- Field Theory and Galois Theory. Field Theory -- Galois theory -- An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra. Commutative rings and algebraic geometry -- Artinian rings, discrete valuation rings, and Dedekind domains -- Introduction to homological algebra and group cohomology -- Introduction to the Representation Theory of Finite Groups. Representation theory and character theory -- Examples and applications of character theory -- Appendices: Cartesian products and Zorn's lemma ; Category theory.