MATHS 311: Algebraic Structures (3)

Syllabus

1.     Prerequisite: Maths 215, 217.

Note: Students with strong performance in Maths 215 and Maths 217 may request permission to substitute Maths 411 for Maths 311.

2.     Course Description:  Consideration of basic algebraic structures: groups, rings, integral domains, and fields. Examples of these structures and elementary proof will be emphasized as will polynomials over rings, integral domains, and the fields of real and complex numbers.

3.     Course Objectives:  Students will learn that algebraic concepts that they have previously encountered are a part of a larger axiomatic algebraic structure including the theory of groups, rings, and polynomials. Students completing this course will not only gain a new insight into “high school” algebra, they will also understand that algebraic concepts are useful throughout higher mathematics. Proof writing will be emphasized; students completing the course should have complete mastery of writing elementary proofs.

4.     Course Rationale: Algebra is a major branch of modern mathematics. Moreover, it has profound connections with other branches of mathematics, including geometry, topology, analysis, and mathematical physics. Furthermore, algebra provides an axiomatic framework in which one performs the algebraic calculations that one might first encounter in high school. Consequently, Maths 311 is an essential course for Mathematics Teaching majors as well as Mathematics majors.

5.     Course Content:  Integers, division algorithm and divisibility, modular arithmetic, prime factorization in the integers. Rings, integral domains, fields (including the complex numbers), homomorphisms. Polynomial rings, divisibility, irreducibility, factorization, connections with the integers. Quotient structures, ideals. Groups, subgroups, Lagrange’s theorem, Cayley’s theorem.

6.     Course Format: Lecture/discussion.

7.     Method of Evaluating Student Performance:  Typical components are homework assignments, projects, and take-home exams (possibly involving group work); in-class exams; student presentations and oral exams. The evaluation and weight of these components are at the discretion of the instructor. Students may be evaluated on their understanding of the content material for the course, their ability to solve problems related to the course material, and on their ability to communicate mathematically to others verbally or in writing.

8.     Evaluation of the Course:  The instruction of the course is evaluated by departmental student evaluations and peer evaluation. The course is reviewed and revised periodically by the Departmental Undergraduate Programs Committee.

R. Bremigan, J. Lorch, 12/8/00

V. Mascioni, 04/07/04

M. Begum, 10/17/05