MATH 301: Abstract Algebra I
(3-0) 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.
Abstract Algebra: An Introduction
- 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.
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
- Students will be able to perform computations involving divisibility of integers.
- Students will be asked to identify ring-theoretic and group-theoretic properties and identify these properties in familiar rings and groups.
- Students will provide proofs to simple assertions of ring- and group-theoretic principles.
Method of Instruction
- Lectures will emphasize ring- and group-theoretic 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
- 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
The course will cover roughly chapters 1-5 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 inN?)
- The well-ordering principle and induction
- 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
- Kernels of functions
- Congruence modulo n
- Addition and multiplication are well-defined
- Integers mod n
- Inverses mod n?
- Arithmetic of C
- Definition (commutative ring with identity)
- Examples and nonexamples: Z, N, Q, R, C, 2Z, 2Z+1, Zn, M2(R), Z[i], continuous functions, polynomials, ring of sets
- Subrings. Reexamine examples
- Fields, zero-divisors, units, integral domains
- Ring homomorphisms
- R[x],R[x1, . . . , xn]
- Degree, coefficients
- Evaluation maps
- 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 in-class 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 hard-copy 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 firstname.lastname@example.org, or by calling 515-294-7220. 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.