# Math 371: Abstract Algebra 1.

## Contents

## Desired Learning Outcomes

This course is aimed at undergraduate mathematics and mathematics education majors. It is a first course in abstract algebra. In addition to being an important branch of mathematics in its own right, abstract algebra is now an essential tool in number theory, geometry, topology, and, to a lesser extent, analysis. Thus it is a core requirement for all mathematics majors. Outside of mathematics, algebra also has applications in cryptography, coding theory, quantum chemistry, and physics.

### Prerequisites

The chief prerequisite for this course is Math 190. In Math 190, students should learn basic logic, basic set theory, the division algorithm, Euclidean algorithm, and unique factorization theorem for integers, equivalence relations, functions, and mathematical induction. As these topics are of high importance in Math 371, it might be prudent for the instructor to review them at the beginning of the semester.

### Minimal learning outcomes

Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements 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 non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

- Group Theory
- Basic Definitions
- Examples of groups
- Subgroups
- Lagrange’s Theorem
- Homomorphisms
- Normal Subgroups
- Quotient Groups
- Isomorphism Theorems
- Cauchy’s Theorem
- Direct Products
- The Symmetric Group
- Even and odd Permutations
- Cycle Decompositions

- Ring Theory
- Basic Definitions
- Examples of rings (both commutative and noncommutative)
- Ideals
- Ring homomorphisms
- Quotient rings
- Prime and maximal ideals
- Polynomial rings
- Factorization in polynomial rings
- Field of fractions of a domain

### Additional topics

Beyond the minimal learning outcomes, instructors are free to cover additional topics. These may include (but are certainly not limited to): Sylow’s theorems, the fundamental theorem of finite abelian groups, the simplicity of *A _{n}*, Burnside’s Theorem, Polya counting, isometries of

**R**

^{3}and regular solids, straight-edge and compass constructions, the 2- and 4-squares theorems, the RSA algorithm, wallpaper groups, coding theory, or latin squares. Instructors are free to use new approaches to the teaching of the material, as long as the minimal learning outcomes are achieved.

### Courses for which this course is prerequisite

This course is required for Math 350, Math 372, Math 387, Math 561, Math 586. As a result it is essential that all required learning objectives be covered.