This continuation of MA511 is designed to give the first year graduate students in mathematics more advanced topics on groups, rings, modules, polynomials, Galois theory, commutative algebra, and multilinear algebra as background for studying number theory and algebraic geometry. Topics include a detailed approach to Galois theory and its applications, transcendental extensions and integral extensions of rings from field theory, simple and semisimple rings, and the representation theory of finite groups.