This is a standard course in second semester graduate algebra for beginning graduate students. Continuing last semester's topics, I will cover:
SField Theory (Field extensions: finite, separable, normal, algebraic and transcendental. Existence of algebraic closure. Galois theory. Finite fields. Hilbert theorem 90); Commutative algebra (Local rings and Nakayama lemma, Integral extensions, Krull dimension, Noether normalization lemma, Hilbert Nullstellensatz, localization. Prime ideal spectrum and Zariski topology, Algebraic sets and rings of regular functions. Discrete valuation rings and Dedekind domains), and Modules (Tensor product, flatness, local properties of modules, exterior and symmetric powers. Graded rings and modules, Hilbert functions and polynomials). If there is time we will also do a little bit of homological algebra.
I'll use Jacobson, (Basic Algebra) as well as Artin (Algebra), and Dummit and Foote (Abstract Algebra).
Standard course in Abstract Algebra for undergraduate students at the level of our Math 451 and Abstract Algebra I