Math 671: Algebra 1.

Catalog Information

Title

Algebra.

Credit Hours

3

Prerequisite

Math 372.

Desired Learning Outcomes

Minimal learning outcomes

Students should achieve an advanced mastery of the topics listed below. This means that they should know all relevant definitions, correct statements and proofs of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts The students should be able to demonstrate their mastery by solving difficult problems related to these concepts, and by proving theorems about the below concepts, even if the theorems go beyond the material in the text.

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
2. 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
3. Module Theory
   • Definitions and Examples
   • Quotient modules and homomorphisms
   • Direct sums
   • Free Modules
   • Tensor Products
   • Exact Sequences
   • Projectives, Injectives, Flats