MTH 281 Introductory Abstract Algebra
Course Title: MTH 281: Introductory Abstract Algebra
Course Description
Introduces groups, isomorphisms, fields, homomorphisms, rings, and integral domains. Applicable to some education licensure programs; not intended for STEM majors. Lecture 3 hours per week. 3 credits.
General Course Purpose
To provide an introduction to abstract mathematics and rigorous proof in the context of algebraic structures to students seeking endorsement to teach mathematics at the secondary level.
Course Prerequisites/Corequisites
Prerequisite: Completion of MTH 263 Calculus I with a grade of C or better or equivalent.
Course Objectives
Upon completing the course, the student will be able to:
Introduction to Logic and Proof

Demonstrate the proof writing strategies of direct proof, indirect proof (proof of contrapositive), and proof by contradiction in the context of proving basic results about integers (e.g. “Prove that the product of two odd integers is odd.”)

State and apply the Well Ordering Principle (for the naturals) and General Well Ordering Principle (for sets of integers bounded below) to conclude whether a given set is guaranteed a smallest element.

Write induction proofs in the context of proving basic results about integers
Operations and Relations

State and apply the definition of an equivalence relation on a set and determine which properties (reflexive, symmetric, transitive) a defined relation on a given set passes or fails.

State and apply the definition of a partial order, and determine if a defined relation on a given set is antisymmetric

Construct equivalence classes given a set and equivalence relation

Determine if a given operation on a set is well defined

Illustrate the construction of integers as equivalence classes of ordered pairs of natural numbers with defined operations of addition and multiplication
Divisibility and Prime Numbers

State and apply the definition of divides and prove basic results about divisibility of integers (e.g. “if ab and bc, then ac”)

Given two integers a and b, apply the Division Algorithm to express a = bq + r, 0 <= r < b

Use the Euclidean Algorithm to find the greatest common divisor of a pair of integers

Obtain integer solutions to Diophantine equations and write the general form of the solution

Write the prime factorization of a given natural number

State and prove the Fundamental Theorem of Arithmetic

State and prove results about prime numbers

Apply the definition of the Euler phifunction to determine the number of numbers relatively prime to a given integer

State and apply Euler’s Theorem and Fermat’s Little Theorem
Modular Arithmetic, Congruence, and an Introduction to Zm

State and apply the definition of congruence modulo m

State and prove fundamental properties of the congruence relation

Perform modular arithmetic on congruence classes of integers

State and prove results about solutions to linear congruences, and apply them to determine solutions

Solve systems of linear congruences

Define Zm and its operations and perform arithmetic within Zm for a given m

Prove that the operations on Zm satisfy the properties of commutativity and associativity of addition and multiplication, and the distributive property of multiplication over addition

Solve linear equations within Zm for a given m
Rings, Fields, and Integral Domains

State the definitions of ring, commutative ring, ring with unity, integral domain, and field

Given a set and two binary operations, determine which of the above structures it falls under by verifying algebraic properties (examples including integer/rational/real/complex numbers with addition and multiplication, Zm rings, and sets and operations in the abstract as defined by Cayley tables)

Determine if an element of a ring is a zero divisor, a unit, or neither

Perform algebraic operations in the complex field, including applying De Moivre’s Theorem to compute powers and nth roots
Polynomials

Perform algebraic operations on polynomials in Q[x] and Z[x] (rational and integer coefficients), and also in Zm[x] (coefficients in a Zm ring with arithmetic modulo m).

Perform long division of polynomials in F[x] (F a field, including Q, Z, C, and Zm, m prime) and express in the form of the Division Algorithm

Use the Euclidean algorithm to find the greatest common divisor of two polynomials in F[x]

State, prove, and apply the Remainder/Root Theorems for polynomials

State and prove the Unique Factorization Theorem for polynomials in F[x]

Determine if a polynomial is reducible in F[x] (apply relevant theorems such as Eisenstein’s Criterion); if so, factor completely

State the Fundamental Theorem of Algebra, and display an understanding of the concepts underlying the proof
Groups, Isomorphism, and Homomorphism

State the definitions of group and Abelian group, and state and prove additional basic properties of groups (e.g. (xy)^1=y^1x^1)

Given a set and a binary operation, determine whether it is a group (and if Abelian) by verifying algebraic properties (examples including integer/rational/real/complex numbers with addition or multiplication, the Klein4 group, and sets and operations in the abstract as defined by Cayley tables)

Construct Cayley tables for the groups Um formed from the units of Zm with the operation of multiplication and perform arithmetic in Um

Define the dihedral groups of symmetries of the triangle and the square and implement operations on elements within the groups

State and apply the definitions of subgroup, proper subgroup, and cyclic subgroup

Construct direct (Cartesian) product groups

State the definition of an isomorphism between two groups and be able to determine if one exists by identifying an operation preserving bijection

State the definition of a homomorphism between two groups and be able to determine if one exists by identifying an operation preserving map
Major Topics to be Included
Introduction to Logic and Proof
Operations and Relations
Divisibility and Prime Numbers
Modular Arithmetic, Congruence, and an Introduction to Zm
Rings, Fields, and Integral Domains
Polynomials
Groups, Isomorphism, and Homomorphism