MATH 6310 Algebra
- Group theory.
- Rings, fields, and modules.
- Introduction to algebraic geometry.
- Multilinear algebra.
I. Group Theory
- Actions of groups on sets; modules; groups with operators.
- Jordan-Hölder theorem in context of groups with operators; simple groups and modules; composition series; solvable groups.
- Orbit formula for action of group on a set; class equation.
- Sylow theorems and consequences.
- Construction and classification of groups of small order; semi-direct product.
- Free groups; generators and relations.
Optional: Nilpotent groups, simplicity of An, wreath product.
II. Rings, Fields, Modules
- Integral domains, fields, quotient fields (brief review, probably without proofs).
- Maximal and prime ideals; existence of maximal left ideals and relation to Zorn’s Lemma.
- Co-maximal (relatively prime) ideals and general Chinese Remainder Theorem.
- Noetherian rings.
- PID’s and UFD’s; examples.
- Polynomial rings, Hilbert’s Basis Theorem, Gauss’s Lemma in some form (in particular, R UFD implies R[x] UFD).
- Finite, algebraic, and primitive field extension; degree formula for field extensions.
- Free modules; structure of modules over PID.
Optional: Localization, Euclidean domains, examples of Dedekind domains and factorization of ideals, algebraic closure, impossibility of ruler and compass constructions.
III. Introduction to Algebraic Geometry
- Algebraic sets and varieties.
- Hilbert’s Nullstellensatz.
- (Wedderburn) radical of commutative ring and ideal; connection with nilpotent elements.
Optional: Jacobson radical, prime and maximal spectrum of commutative ring.
IV. Multilinear Algebra
- Universal properties.
- Tensor product of modules.
- Tensor algebra of module.
- Exterior algebra of module over commutative ring.
Optional: Graded rings, tensor product of bimodule, symmetric algebra, Lie and enveloping algebra, Clifford algebra, Weyl algebra.