MATH 6310 Algebra

Main Topics

  1. Group theory.
  2. Rings, fields, and modules.
  3. Introduction to algebraic geometry.
  4. Multilinear algebra.


I. Group Theory

  1. Actions of groups on sets; modules; groups with operators.
  2. Jordan-Hölder theorem in context of groups with operators; simple groups and modules; composition series; solvable groups.
  3. Orbit formula for action of group on a set; class equation.
  4. p-Groups
  5. Sylow theorems and consequences.
  6. Construction and classification of groups of small order; semi-direct product.
  7. Free groups; generators and relations.

Optional:  Nilpotent groups, simplicity of An, wreath product.

II. Rings, Fields, Modules

  1. Integral domains, fields, quotient fields (brief review, probably without proofs).
  2. Maximal and prime ideals; existence of maximal left ideals and relation to Zorn’s Lemma.
  3. Co-maximal (relatively prime) ideals and general Chinese Remainder Theorem.
  4. Noetherian rings.
  5. PID’s and UFD’s; examples.
  6. Polynomial rings, Hilbert’s Basis Theorem, Gauss’s Lemma in some form (in particular, R UFD implies R[x] UFD).
  7. Finite, algebraic, and primitive field extension; degree formula for field extensions.
  8. 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

  1. Algebraic sets and varieties.
  2. Hilbert’s Nullstellensatz.
  3. (Wedderburn) radical of commutative ring and ideal; connection with nilpotent elements.

Optional: Jacobson radical, prime and maximal spectrum of commutative ring.

IV. Multilinear Algebra

  1. Universal properties.
  2. Tensor product of modules.
  3. Tensor algebra of module.
  4. Exterior algebra of module over commutative ring.

Optional: Graded rings, tensor product of bimodule, symmetric algebra, Lie and enveloping algebra, Clifford algebra, Weyl algebra.