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Master Syllabus

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Administrative Unit: Computer and Mathematical Sciences Department
Course Prefix and Number: MATH 304
Course Title: Introduction to Abstract Algebra
Number of:
Credit Hours 3
Lecture Hours 3
Lab Hours 0
Catalog Description: 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.
Text(s): Most current editions of the following:

A First Course in Abstract Algebra
By Fraleigh (Addison Wesley)
Abstract Algebra: An Introduction
By Hungerford (Harcourt)
Course Objectives
  • To describe and explain the theoretical foundations of abstract algebra, including group, ring and field theory.
  • To recognize valid mathematical arguments.
  • To verify precise mathematical definitions as one step in proving mathematical theorems.
  • To provide examples and counterexamples for mathematical statements.
  • To communicate mathematically, formally and informally, both verbally and in writing.
    Measurable Learning
  • 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.
    Topical Outline:
  • A review of sets and relations
  • Groups and subgroups
    - Introduction
    - Isomorphic binary pperations
    - Groups
    - Subgroups
    - Cyclic groups
  • Permutations, cosets, and direct products
  • Groups of permutations
  • Orbits, cycles, and the alternating group
  • Cosets and the Theorem of Language
  • Direct products and finitely generated Abelian groups
  • Homomorphisms and factor groups
  • Homomorphisms
  • Factor groups
  • Factor-group computations and simple groups
  • Rings and fields
  • Rings
  • Fields
  • Integral domains
  • Fermat’s and Euler’s Theorems
  • The field of quotients of an integral domain
  • Rings of polynomials
  • Factorization of polynomials over a field
  • Ideals and factor rings
  • Homomorphisms of factor rings
  • Prime and maximal ideals
    Culminating Experience Statement:

    Material from this course may be tested on the Major Field Test (MFT) administered during the Culminating Experience course for the degree. 
    During this course the ETS Proficiency Profile may be administered.  This 40-minute standardized test measures learning in general education courses.  The results of the tests are used by faculty to improve the general education curriculum at the College.


    Recommended maximum class size for this course: 20

    Library Resources:

    Online databases are available at You may access them from off-campus using your CougarTrack login and password when prompted.

    Prepared by: Ann Schlemper Date: November 7, 2013
    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 objectives and cover the subjects listed in the topical outline. 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.

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