# Difference between revisions of "Math 371: Abstract Algebra 1."

### Title

Abstract Algebra.

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F, W, Sp

### Description

Groups, rings, fields, vector spaces, linear transformations, matrices, field extensions, etc.

## 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.

1. 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
2. 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