MA312 ABSTRACT ALGEBRA I (3 Cr.)
Prerequisite: MA 211 (Introduction to Matrix Theory ) and MA 163 (Calculus II), or permission of instructor
Topics chosen from the following: universal algebraic notions, graphs, trees, lattices, Boolean algebras, groups, rings, fields; applications.
Goal/Purpose of the Course
MA 312 introduces students to the concepts of methodology of abstract algebra. The basic definitions and fundamentals theory of groups, rings, integral domains and field are explored, and the role of homomorphisms in algebra is discussed. Attention is given to developing the students' skills in using the notation of the predicate calculus as a guide for rigorous reasoning. The applicability of the organized approach taken in abstract algebra is illustrated for the students by means of solutions to several problems which apply ideas from abstract algebra to other parts of mathematics or to problems arising in computer science or other areas.
Objectives of the Course
Upon completion of MA 312, students should be able to:
- state definitions of groups, rings, integral domains and fields.
- give a variety of examples of these algebraic structures.
- demonstrate familiarity with the basic theory of groups and rings.
- develop rigorous proofs or basic algebraic results.
- write proofs and mathematical exposition using good style.
- address successfully open-ended questions by exploring the relevant issues, formulating specific questions, identifying relevant examples or counter-examples, formulating conjectures and proving or disproving them, and clearly writing up their results.
Course Content Outline
- Fundamental Concepts
- Relations and their properties
- Formal notation: The Predicate Calculus
- Group Theory
- Definition and simple properties
- Groups of permutations
- Symmetric groups on n symbols
- Cyclic groups
- Cosets and Lagrange's theorem
- Quotient groups
- The fundamental homomorphism theorem
- Ring Theory
- Definitions and simple properties of rings, domains and fields
- Properties of the integers
- the division algorithm
- greatest common divisor and least common multiples
- order properties
- the principle of mathematical induction and well ordering
- characterization of the ordered domain of integers
- Finite fields and Boolean rings
- Ideals and quotient rings
- Characteristic of a ring
- Methods of constructing new rings
- fields of quotients of integral domains
- adjoining a unity to a ring
- direct sums of rings
- rings of matrices and their subrings
- Using symmetric groups to solve Rubik's Cube and related puzzles
- Using the projective plane over a field with four elements to solve a tournament scheduling problem
- Boolean rings as a tool for logic and computing