# Math 671: Algebra 1.

Algebra 1.

3

### Prerequisite

Math 372 or equivalent.

## Desired Learning Outcomes

### Prerequisites

Math 372 is a prerequisite for this course. In particular, students should be familiar with the concepts of groups and rings, and they should understand constructions of quotient groups and quotient rings. By this point in their mathematical career, students should be skilled at proving theorems by themselves.

### 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

### Textbooks

Possible textbooks for this course include (but are not limited to):