Proposed Syllabus for Math 566/567

Tested topics for M566 are in blue.  Tested topics for M567 are in red.

Groups

Basic definitions.  Subgroups and cosets.  Lagrange's Theorem.
The symmetric/alternating group (cycle structure, conjugacy classes) and other examples of groups.
Direct products.

Homomorphisms and Quotient Groups

Homomorphisms, image, kernel.  Normal subgroups.  Quotient groups.
Isomorphism Theorems. Centralizers, Normalizers.

Group Actions

Groups acting on sets.  Stabilizers, orbits.  Counting lemmas.  Cayley's Theorem.
Semi-direct products.  The class equation.
The Sylow Theorems;  applications to classification of finite groups.

Rings

Basic definitions.  Polynomial rings, matrix rings, group rings.
Ideals and quotient rings.  Maximal ideals.
Homomorphisms.  Isomorphism theorems.

Ideals in commutative rings.

Modules

Basic definitions and examples.
Submodules, quotients, homomorphisms, isomorphism theorems.
Module constructions: direct sum, generation, free modules, torsion.

Finitely Generated Modules over Principal Ideal Domains

Classification Theorem.  Invariant factors, elementary divisors.
Applications:  classification of finitely generated abelian groups, rational canonical form, Jordan form.

Fields

Field extensions: degree, algebraic and transcendental elements, transcendence degree.
Finite fields.
Algebraic extensions, separability, splitting fields.   Cyclotomic fields.  Algebraically closed fields.

Galois Theory

Galois extensions, the Galois correspondence.
Galois groups of polynomials.
Solvable extensions and applications.

Some optional topics

             Grobner basis techniques for polynomial rings
             Rings and modules of fractions
             Jordan/Holder theory, composition series
             Hom and tensor functors;  projective and injective objects
             Symmetric and exterior algebras
             Basic representation theory