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 in N?)
- 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%