Course Coordinator
Jason McCullough
Catalog Description
MATH 301: Abstract Algebra I
(30) Cr. 3. F.S.
Prereq: MATH 166 or MATH 166H, MATH 317, and grade of C or better in MATH 201
Basic properties of integers, divisibility and unique factorization. Polynomial rings over a field. Congruence. Introduction to abstract rings, homomorphisms, ideals. Roots and irreducibility of polynomials. Introduction to groups. Emphasis on proofs.
Textbook
Abstract Algebra: An Introduction
Hungerford
ISBN: 9781111569624
Prerequisites
 Math 166 (Calculus II), Math 317 or 407 (Linear Algebra), and Math 201 (Introduction to Proofs)
 A student who has taken Math 207 in lieu of 317 may be prepared for the course. Discuss with the instructor. While the construction of sound proofs will be a central component of the course, a student with no previous experience writing simple proofs may find the course overly challenging.
Learning Outcomes
Upon completion of this course, students…
 Will be familiar with properties of the integers such as prime factorization, divisibility, and congruence
 will be able to reason abstractly about mathematical structures
 will recognize and comprehend correct proofs of formal statements and be able to formulate proofs clearly and concisely
Learning Objectives
 Students will be able to perform computations involving divisibility of integers.
 Students will be asked to identify ringtheoretic and grouptheoretic properties and identify these properties in familiar rings and groups.
 Students will provide proofs to simple assertions of ring and grouptheoretic principles.
Method of Instruction
 Lectures will emphasize ring and grouptheoretic properties. Weekly homework assignments will ask students to recognize these properties.
 Weekly homework assignments.
 Numerous proofs will be presented in class. Students will construct proofs on weekly homework assignments
Assessment
 Exam question: Which of the following rings is an integral domain…
 Exam question: Write the permutation as a product of disjoint cycles. Does the permutation lie in the alternating group?
 Exam question: Prove the following assertion about rings…
Determine the proportion of students answering the questions correctly
Course Topics
The course will cover roughly chapters 15 and 7 of the text. A list of topics is below.
 The integers and other familiar sets of numbers
 N,Z,Q,R,C (Should we include 0 in N?)
 The wellordering principle and induction
 Divisibility
 The division algorithm
 greatest common divisors. Euclidian algorithm?
 Bezout’s identity
 Least common multiples
 Prime numbers
 Euclid’s lemma
 Fundamental theorem of arithmetic
 Infinitely many primes
 Functions, Domain, Range, etc.
 Equivalence Relations
 Partitions
 Kernels of functions
 Welldefinededness
 Congruence modulo n
 Addition and multiplication are welldefined
 Integers mod n
 Inverses mod n?
 Arithmetic of C
 Rings
 Definition (commutative ring with identity)
 Examples and nonexamples: Z, N, Q, R, C, 2Z, 2Z+1, Z_{n}, M_{2}(R), Z[i], continuous functions, polynomials, ring of sets
 Subrings. Reexamine examples
 Fields, zerodivisors, units, integral domains
 Ring homomorphisms
 Polynomials
 R[x],R[x1, . . . , xn]
 Degree, coefficients
 Evaluation maps
 Roots
 F[x]/(p(x))
 Groups
 Definition of group, abelian group
 Motivating examples. Additive and multiplicative structure of a ring.
 Simple computations involving elements. Order of an element.
 Subgroups. Cyclic subgroups.
 Group homomorphisms
 Symmetric and alternating groups. Cycle decomposition.
Exams and Grading
The class will require weekly homework submissions, 2 inclass exams, and a final exam. Exam dates and content will be determined by the individual instructor. Class components will be weighted as follows.

Homework – 45%

Exam 1 – 15%

Exam 2 – 15%

Final Exam – 25%
Free Expression Statement
Iowa State University supports and upholds the First Amendment protection of freedom of speech and the principle of academic freedom in order to foster a learning environment where open inquiry and the vigorous debate of a diversity of ideas are encouraged. Students will not be penalized for the content or viewpoints of their speech as long as student expression in a class context is germane to the subject matter of the class and conveyed in an appropriate manner.
Students With Disabilities
Iowa State University is committed to assuring that all educational activities are free from discrimination and harassment based on disability status. Students requesting accommodations for a documented disability are required to work directly with staff in Student Accessibility Services (SAS) to establish eligibility and learn about related processes before accommodations will be identified. After eligibility is established, SAS staff will create and issue a Notification Letter for each course listing approved reasonable accommodations. This document will be made available to the student and instructor either electronically or in hardcopy every semester. Students and instructors are encouraged to review contents of the Notification Letters as early in the semester as possible to identify a specific, timely plan to deliver/receive the indicated accommodations. Reasonable accommodations are not retroactive in nature and are not intended to be an unfair advantage. Additional information or assistance is available online at www.sas.dso.iastate.edu, by contacting SAS staff by email at accessibility@iastate.edu, or by calling 5152947220. Student Accessibility Services is a unit in the Dean of Students Office located at 1076 Student Services Building.
More information about disability resources in the Mathematics Department can be found at https://iastate.app.box.com/s/c17d3ljul83lujr2j1mdeqoqcdqiva1t.