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**NVCC COLLEGE-WIDE COURSE CONTENT SUMMARY**

**MTH 200 – ABSTRACT ALGEBRA (3 CR.)**

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**Catalog
description**

Prerequisite: MTH 174 or permission of instructor.** **Topics covered include groups,
isomorphisms, fields, homomorphisms, rings, and integral domains. Designed to
fulfill the abstract algebra requirement for the Virginia high school
mathematics teaching endorsement.
Lecture 3 hours per week. 3 credit hours.

**Course
Description:**

Presents topics in abstract algebra to
fulfill the abstract algebra requirement for the Virginia high school
mathematics teaching endorsement. Studies groups, isomorphisms, fields,
homomorphisms, rings, and integral domains. Lecture 3 hours per week.

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**General
course purpose**

This course is designed to fulfill
Virginia's requirement for an abstract algebra course as part of the State's
certification requirement for teaching mathematics at the secondary school
level. The course should give the student understanding of the following:

A. the
required pieces of an abstract algebra system

B. the importance and use of abstract
algebraic structures in mathematics

C. the
importance and use of isomorphisms

D. the origin and use of __field
properties__ in a high school algebra course

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**Entry
level-competencies**

MTH
174 - "Calculus with Analytic Geometry II" or consent of the
division.

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**Course objectives**

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As
a result of the learning experiences provided in this course, the student
should be able to:

A. define the basic terms of abstract
algebra:

group, subgroup,
isomorphism, normal subgroup, homomorphism, ring, integral domain, and field.

B.
determine whether a given
structure is a group, a field, or a ring, as appropriate

C.
given two groups, determine if
they are isomorphic, homomorphic, or if one is a subgroup of the other.

D.
illustrate group behaviors using
finite groups, cyclic groups, and permutation groups

E.
illustrate ring, field, or
integral domain behaviors using appropriate models

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**Major
topics to be included**

A. groups

1. definition

2. subgroups

3. normal
subgroup

4. types
of groups

a. cyclic

b. finite

c. permutation

d. direct product

e. dihedral group

B. Isomorphism and homomorphism

1. definition

2. properties

3. Cayley’s
Theorem

4. factor
groups

C. Rings, Fields, and Integral Domains

1. definitions

2. properties
of each

3. examples
of each