Catalog Description

MATH 305 Abstract Algebra I (3-0-3)(S) Introduction to abstract algebraic systems their motivation, definitions, and basic properties. Primary emphasis is on group theory (permutation and cyclic groups, subgroups, homomorphism, quotient groups) and the theory of finite fields, followed by a brief survey of rings, integral domains, and fields.
PREREQ: MATH 175 and MATH 187.

Prerequisites

MATH 175 Calculus and Analytic Geometry and M 187 Discrete and Foundational Math, or equivalent courses elsewhere, or permission of the instructor. The rationale for these prerequisites is that Abstract Algebra requires a level of Mathematical maturity which could be attained through these two lower division courses.

Jurisdiction

This course is not currently controlled by a departmental committee and individual instructors may choose different textbooks. Exams, homework, and grading system are left to the instructor.

Learning Objectives

As a course in Mathematics, the objectives of MATH 305 include that students be able to:

use results from elementary group theory to solve contemporary problems;
explain from elementary principles why certain algebraic facts are true;
use available computer programs to solve problems from the genre of computational group theory.

Currently this course is taken mainly by mathematics majors.

Upon completion of this course, students should be able to:

explain the fundamental concepts of finite group theory and finite field theory;
use available software to do the computational part of problem solving;
use Lagrange's Theorem to analyse the cyclic subgroups of a group;
explain the significance of the notion of a normal subgroup, and of a simple group;
describe the structure of finite Abelian groups;
use Sylow's theorems to describe the structure of certain finite groups;
explain the notion of an extension of a field;
describe the structure of finite fields;
do computations in specific examples of finite fields.

Assessment of Learning Objectives

Students will be assessed by evaluating their ability to do problems based on the learning objectives. The problems may occur in several contexts:

Periodic problem sets for homework serve both as learning and as assessment tools. Classroom activities may vary depending on students' outcomes on homework assignments.
Problems given on take-home exams are designed to evaluate a student's ability to solve more complicated and time consuming problems. Moreover, take-home problems give students the opportunity to demonstrate their ability to use technology to solve problems involving real data or otherwise not amenable to simple analytic techniques.
Problems given on in-class exams are designed to give students the opportunity to demonstrate their ability to work simpler and less time consuming problems which probably do not involve real data.

Topics and Approximate Timeline

The following table is based on a typical semester schedule-60 class meetings of 50 minutes each. The actual amount of time spent on each topic will vary slightly from semester to semester and instructor to instructor.


MATH 305 Abstract Algebra I
	Number of
Topic	Meetings
Geometry and groups	3
Definition and examples of groups	3
Finite groups and criteria for subgroups	3
Cyclic groups	3
Permutation groups	6
Cosets and Lagrange's theorem	4
Direct products	6
Normal subgroups and group homomorphisms	6
The structure of finite Abelian groups	6
Sylow's theorems	6
Rings and Fields	2
Finite Fields	8
Exams	3
Total	59

Text

The current (Spring 1999) text is Contemporary Abstract Algebra, Joseph A. Gallian, Houghton Mifflin Company (1998) plus Laboratory Experiences in Group Theory, E.M. Parker, Mathematical Association of America (1996). Another textbook that has been used in recent years is Abstract Algebra, I.N. Herstein, Prentice-Hall (1996).

Web sites

The following websites offer valuable learning materials for this course:

http://www.d.umn.edu/ jgallian Computer Software for Abstract Algebra
http://euler.slu.edu/courseware/AbstractAlgebra/AlgebraOverview.html Maple Worksheets for Abstract Algebra
http://unr.edu/homepage/keppelma/fgb.html Finite Group Behavior
http://www.maths.usyd.edu.au:8000/u/magma/ The Magma Computational Algebra System

Format, Student Activities, and Grades

Class meetings involve a combination of lecture, questions and discussion, and sometimes small group activity; the instructor chooses the appropriate mix. The computer algebra system, Maple, is used for laboratory activities and homework. Homework is an important part of the course; many exercises involve extensions of ideas in the text to new situations, rather than just routine applications. Some exams may be partially take-home. The instructor chooses the exact grading scheme, but a typical distribution might be:


Homework (including a writing assignment and project)	33 ¹/₃ %
Exams	33 ¹/₃%
Final Exam	33 ¹/₃%
Total	100%

Letter grades are based on a scale in which 90% of the total possible points guarantees an A, 80% a B, 70% a C, and 60% a D, with the instructor having the discretion to lower these cut-offs if warranted.

File translated from T_EX by T_TH, version 1.56.