Introduction to algebraic systems, their motivation, definitions and basic properties. Primary emphasis is on group theory (permutation and cyclic groups, subgroups, homomorphism, quotient groups) and is followed by a brief survey of rings, integral domains and fields. Prerequisites: Grade of C or higher in both MATH 222 and MATH 225.
Prerequisite(s) / Corequisite(s):
Grade of C or higher in both MATH 222 and MATH 225.
Course Rotation for Day Program:
Offered even Spring.
Most current editions of the following:
A First Course in Abstract Algebra
By Fraleigh (Addison Wesley) Recommended
Abstract Algebra: An Introduction
By Hungerford (Harcourt) Recommended
Course Learning Outcomes
Describe and generate groups, rings, and fields.
Relate abstract algebraic constructs to more familiar number sets and operations and see from where the constructs derive.
Identify examples of specific constructs.
Identify and differentiate between different structures and understand how changing properties give rise to new structures.
Explain the theory behind relations and functions and identify domains and images of functions, based on the structures given.
Explain how functions may relate seemingly dissimilar structures to each other and how knowing properties of one structure allows us to know the same properties in the related structure, if certain functions exist between them.
Major Topics/Skills to be Covered:
Define and use operations and sets.
State the definition of a group and give examples.
State, prove and use elementary theorems of groups.
Define and find subgroups of groups.
State, prove and use basic theorems of subgroups.
State and prove theorems regarding cycle groups, permutation groups and isomorphisms.
Find cosets and prove theorems regarding cosets.
Determine whether a given map is a homomorphism.
Compute the kernel and range of a homomorphism.
State and apply properties of normal subgroups.
Prove elementary properties of homomorphism.
Define factor groups, give examples and apply properties of factor groups.
State and use the fundamental homomorphism theorem.
Define and give examples of rings, integral domains and fields.
State, prove and use elementary properties of rings, integral domains and fields.
Define and give examples of ideals, ring homomorphisms and factor rings.
State and use the fundamental homomorphism theorem for rings and some consequences.
State and use the characteristics of integral domains and ideals.
Recommended maximum class size for this course: 30
NOTE: The intention of this master course syllabus is to provide an outline of the contents of this course, as specified by
the faculty of Columbia College, regardless of who teaches the course, when it is taught, or where it is taught. Faculty members teaching this
course for Columbia College are expected to facilitate learning pursuant to the course learning outcomes and cover the subjects listed in the Major Topics/Skills to be Covered section.
However, instructors are also encouraged to cover additional topics of interest so long as those topics are relevant to the course's
subject. The master syllabus is, therefore, prescriptive in nature but also allows for a diversity of individual approaches to course material.