Algebra I

Department: 
MATH
Course Number: 
6121
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every fall semester

Graduate level linear and abstract algebra including groups, rings, modules, and fields. (1st of two courses)

Prerequisites: 

MATH 4107 and one of MATH 2406, MATH 4305, or permission of instructor

Course Text: 

Text at the level of Abstract Algebra by Dummit and Foote.

Topic Outline: 

  1. Groups: Basic facts about groups, including cyclic, dihedral, symmetric, and linear (matrix) groups; homomorphisms; cosets and quotients; normal subgroups; mapping properties and isomorphism theorems; group actions; applications of group actions to geometric and combinatorial symmetry (e.g. symmetries of regular polyhedra, Polya enumeration, and Burnside's formula); the class equation; the Sylow theorems; simple groups and composition series; the structure theorem for finitely generated abelian groups.

  2. Rings: (By a ring we mean a commutative ring with identity.) Definitions; homomorphisms; ideals; quotients; mapping properties and isomorphism theorems; polynomial rings; integral domains; fraction fields; prime and maximal ideals; Euclidean domains; unique factorization domains; principal ideal domains; Gauss's lemma; irreducibility criteria; the Chinese remainder theorem.

  3. Modules: Definitions and examples; homomorphisms and quotients; generation, freeness, and finiteness properties; the structure theorem for modules over a principal ideal domain.

  4. Fields: Characteristic; prime fields; field extensions; algebraic vs. transcendental extensions; splitting fields; basic properties of finite fields; algebraic closure.

  5. Galois Theory: Definitions; the fundamental theorem of Galois theory; examples, including quadratic, cubic, cyclotomic, and finite fields; the primitive element theorem.