MTH601: Topics in algebra

[Cr:4, Lc:3, Tt:1, Lb:0]

Groups acting on sets, Quotients, Direct products, Semidirect products, Exact Sequences, Automorphism groups of various objects, Linear groups, Nilpotent and Solvable groups, Structure of finitely generated abelian groups, Sylow Theorems, Free Groups, Generators and relations, Amalgamation.
Linear Transformations, Eigenvalues and eigenvectors, Triangulation, Diagonalizable transformations, Spectral Theorems.
Modules, Direct Products and Direct Sums, Modules over PIDs, Various Canonical Forms, Simple and semisimple modules, Semisimple rings, Wedderburn-Artin structure Theory.
Categories, Groupoids, Small categories, Full subcategories, Functors, Left and right adjoints of functors, Universal objects, Natural transformations of functors, Equivalence of categories.
Algebraic numbers, Field extensions, Constructible numbers, Ruler and Compass Constructions, Splitting Fields, Algebraic Closure and Normality, The Fundamental Theorem of Algebra, Separability, Galois extensions, Galois group of a Galois extension, The Galois Pairing, Finite Fields, Cyclic Extensions, Cyclotomic Extentions, Solvability by radicals, The general equation of degree , Transcendental extensions and transcendence basis.