# MAT 333 Abstract Algebra

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### Course Description

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

3 units credit.

### Prerequisites

MAT 271 or equivalent with a grade of "C" or better.

### Text

Texts are chosen by the instructor. For example:

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

### Course Requirements, Tentative Schedule of Class Meetings and Topics, Readings, Assignments and Due dates, Exams

A schedule of class meetings, topics, assignments, due dates, exam dates, etc. will be provided by instructor. See your class syllabus.

Here is an example course outline, based on the above text.

• 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

The final exam is given at the date and time announced in the Schedule of Classes.

### Learning Objectives

After completing MAT 333 the student will

• state definitions of basic concepts (e.g., congruence, groups, rings, integral domains, fields, subrings, homomorphism, ideal)
• explain and use the Euclidean algorithm
• explain 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.

### Computers and Calculators, Computer Literacy

Most instructors encourage the use of machines, calculators computers, phones etc., for analyzing data. The use of machines may be restricted during examinations or at certain other times. Ask your instructor for the policy in your class.

Students are not expected to be programmers or to know any particular computer language before starting this class. Some instructors may expect students to be able to access information on the internet, or to use calculators, or to learn to use particular software with instruction. Basic skill in algebra and the use of mathematical symbols, order of operations etc., and the willingness to read and follow instruction manuals and help files will suffice.

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in this course. The instructor determines the relative weights of these factors and the grading scale. See the syllabus for your particular class.

### Location of Class Meetings

Classes meet on the dates and room announced in the official Schedule of Classes. This is a traditional, face-to-face class.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates, Make-Up Work, Missed Exams, and Extra-Credit Assignments

Due dates and policy regarding make-up work and missed exams are set by the instructor. Instructors may, or may not, choose to offer extra credit assignments. If extra credit assignments are offered they will be available to all students.

The mathematics department does not tolerate cheating. Students who have questions or concerns about academic integrity should ask their professors or the counselors in the Student Development Office, or refer to the University Catalog for more information. (Look in the index under "academic integrity".)

### Accomodations for Students with Disabilities

Cal State Dominguez Hills adheres to all applicable federal, state, and local laws, regulations, and guidelines with respect to providing reasonable accommodations for students with temporary and permanent disabilities. If you have a disability that may adversely affect your work in this class, I encourage you to register with Disabled Student Services (DSS) and to talk with me about how we best can help you. All disclosures of disabilities will be kept strictly confidential. Please note: you must register with DSS to arrange an no accommodation. For information call (310) 243-3660 or send an email message to dss@csudh.edu or visit the DSS website http://www4.csudh.edu/dss/contact-us/index or visit their office WH D-180

### Behavioral Expectations

We all are adults so behavior rarely is an issue. Just follow the Golden Rule: "do unto others as you would have them do unto you" then everything will be fine.

The university must maintain a classroom environment that is suitable for learning, so anyone who insists on disrupting that environment will be expelled from the class.

Revision history:

Prepared by J. Barab 2/04/00. Revised 4/28/01, 7/25/06, 1/10/15 (G. Jennings).