Syllabus for the First Year Graduate Algebra (504-506), 2018-2019
(tentative, some topics are likely to spill over to other quarters):

 

Fall quarter  

·         Group theory.  Normal and factor groups, homomorphisms, conjugacy, isomorphism theorems, p-groups and Sylow theorems.  Nilpotent and solvable groups, normal series, commutator subgroup, free groups, simple groups, Jordan-Holder theorem. Direct and semi-direct products, extensions, free product.

·         Category theory. Categories and functors, natural transformations, universal properties, products and coproducts.

·         Ring theory, commutative case. Ideals and homomorphisms, factor rings, maximal and prime ideals, Zorn’s lemma, Eucledian rings, PID and UFD, Gauss lemma. Polynomial rings, elementary symmetric polynomials.

·         Field theory. Field extensions: finite, separable, normal, algebraic and transcendental. Finite fields, algebraic closure, Galois theory.

·         Matrix theory (mostly independent review). Characteristic and minimal polynomial, Cayley-Hamilton theorem, canonical Jordan form.

 

Winter Quarter

·         Field theory, continued.  Kummer and cyclotomic extensions, solvability by radicals. Kummer theory.

·         Ring theory, commutative case, continued. Rings and modules. Structure theory for PID, finiteness conditions, Noetherian rings and modules, Krull dimension, integrality, going up and down theorems, Hilbert basis theorem.

·         Ring theory, non-commutative case. Artinian rings, group rings, semi-simple rings, irreducible and indecomposable modules, Artin-Wedderburn theorem.

·         Group actions and representation theory. Group algebras, irreducible representations, Schur lemma, complete reducibility of representations, character theory.

 

Spring Quarter

·         The main emphasis will be on commutative algebra with strong geometric flavor.  We shall cover Hilbert’s Nullstellesatz, prime ideal spectrum and Zariski topology, localization, tensor product, flatness, exterior and symmetric powers, discrete valuations and Dedekind rings, graded rings and modules, Hilbert functions and polynomials.

·         Very basic introduction to homological algebra: projective and injective modules, resolutions, chain complexes, (left and right) exact functors, derived functors, Tor and Ext.

 

References:

 

General

o   Algebra, S. Lang

o   Algebra, T. Hungerford

o   Abstract Algebra, D. Dummit and R. Foote

o   Basic Algebra II, N. Jacobson

o   Algebra, M. Artin (more of an undergraduate text)

 

Group theory and representations

o   An Introduction to the Theory of Groups, Rotman

o   Representation theory: A first course, Fulton and Harris

o   Linear Representations of Finite Groups, J.-P. Serre

o   Algebras and Representation Theory; K. Erdmann, T. Holm

 

Homological algebra  

o   Introduction to homological algebra, C. Weibel

 

Commutative algebra

o   Introduction to Commutative Algebra, M. Atiyah, I. Macdonald

o   Commutative Algebra for Undergraduates, M. Reid

o   Algebraic Geometry for Undergraduates, M. Reid