# MAT 333 Abstract Algebra

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### Course Description

The theory of groups, rings, ideals, integral domains, fields and related results.

MAT 333 meets for three hours of lecture per week.

### Prerequisites

Students must have earned at least a "C" in a course equivalent to MAT 271-Foundations of Higher Mathematics before enrolling in this course. A semester of linear algebra (MAT 331) is recommended but not required.

### Objectives

After completing MAT 333 the student should be able to

• state definitions of basic concepts (e.g., congruence, groups, rings, integral domains, fields, subrings, homomorphism, ideal)
• understand and use the Euclidean algorithm
• understand and use modular arithmetic and the Chinese remainder theorem
• state major theorems (e.g., the division algorithm, the unique factorization theorem, the remainder theorem, the factor theorem, the first, second, and third isomorphism theorems, classification of cyclic groups, Cayley's theorem) and be able to identify the structures to which each theorem applies (e.g. the integers, integral domains, polynomial rings k[x] where k is a field, groups, etc.)
• find examples of objects that satisfy given algebraic properties (a noncommutative ring, a commutative ring but not an integral domain, etc)
• determine whether a given conjecture is true or false, then prove or disprove it, constructing examples where appropriate
• prove that two rings or groups are, or are not, isomorphic
• prove that a given set is, or is not, a subring (subgroup, subfield, ...) of a given ring (group, field,...)
• prove that congruence classes (or cosets) in a set S do, or do not, inherit given properties from S
• write proofs of other simple propositions using basic definitions and theorems
• use the techniques of abstract algebra to solve applied problems, as appropriate.

### Expected outcomes

Students should be able to demonstrate through written assignments, tests, and/or oral presentations, that they have achieved the objectives of MAT 333.

### Method of Evaluating Outcomes

Evaluations are based on homework, class participation, short tests and scheduled examinations covering students' understanding of topics that are covered in MAT 333.

### Text

Abstract Algebra, An Introduction (2nd edition), by Thomas W. Hungerford.

#### Suggested List of Topics

• Arithmetic in the Integers
• The Division algorithm
• Divisibility
• Primes and Unique Factorization
• Congruence in the Integers and Modular Arithmetic
• Congruence and Congruence Classes
• Modular arithmetic
• Rings
• Definitions, Examples, and Properties of Rings
• Subrings, Integral Domains, and Fields
• Isomorphisms and Homomorphisms
• Arithmetic in Polynomial Rings F[x]
• Division Algorithm and Divisibility in F[x]
• Irreducibles and Unique Factorization
• Polynomial Functions, Roots, and Reducibility
• Irreducibility in Polynomial Rings over the Rationals, Reals, and Complex Fields
• Congruence in F[x]
• Congruence-Class Arithmetic in F[x]
• The Structure of F[x]/(p(x)) when p(x) is Irreducible
• Ideals and Quotient Rings
• Ideals and Congruence
• Quotient Rings and Homomorphisms
• The Structure of R/I when I is Prime or Maximal
• Groups
• Definitions, Examples, and Properties of Groups
• Subgroups
• Isomorphisms and Homomorphisms
• Additional Topics as time permits

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in MAT 333. The instructor determines the relative weights of these factors.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates and Make-Up Work

Due dates and policy regarding make-up work are set by the instructor.

### Schedule of Examinations

The instructor sets all test dates except the date of the final exam. The final exam is given at the date and time announced in the Schedule of Classes.