Math 671: Algebra 1.

Catalog Information

Title

Algebra.

Credit Hours

3

Prerequisite

Math 372.

Desired Learning Outcomes

Minimal learning outcomes

1. Group Theory
   • Axioms and Examples
   • Homomorphisms and Isomorphisms
   • Subgroups
   • Centralizers and Normalizers
   • Cyclic gorups and subgroups
   • Quotient Groups
   • Lagrange's Theorem
   • Isomorphism theorems
   • Group Actions
   • Permutation Representations
   • Cayley's Theorem
   • The class equation
   • Sylow theorems
   • Direct and semidirect products
   • Solvable and Nilpotent groups

1. Ring Theory
   • Definitions and Examples
   • Homomorphisms and quotient rings
   • Ideals
   • Rings of fractions
   • Chinese remainder theorem
   • Euclidean Domains, PID's and UFD's
   • Polynomial Rings

1. Module Theory
   • Definitions and Examples
   • Quotient modules and homomorphisms
   • Direct sums
   • Free Modules
   • Tensor Products
   • Exact Sequences
   • Projectives, Injectives, Flats