Fall 2022
- MATH 415-502: Modern Algebra I
Time and venue: TR 2:20–3:35 p.m., BLOC 163
Office hours (BLOC 301b):
- TR 11:00 a.m.–12:00 p.m.
- by appointment
Office hours (ZOOM meeting):
- Wednesday 5:00–6:00 p.m.
- by appointment
Office hours during the finals (ZOOM meeting):
- Friday, December 9, 5:00–6:00 p.m.
- Monday, December 12, 5:00–6:00 p.m.
- by appointment
Office hours during the finals (BLOC 301b):
- Tuesday, December 13, 11:00 a.m.–1:00 p.m.
- Wednesday, December 14, 10:30 a.m.–12:30 p.m.
Exam 2: Thursday, November 10 (Sample problems)
Final exam: Wednesday, December 14, 1:00-3:00 p.m. (Sample problems)
Course outline:
Part I: Basic group theory
- Preliminaries from set theory
- Binary operations
- Groups, semigroups
- Subgroups, cyclic groups
- Groups of permutations
- Cosets, Lagrange's theorem
Fraleigh/Brand: Chapters I and II
Lecture 1: Preliminaries from set theory. Cardinality of a set.
Lecture 2: Cardinality of a set (continued). Binary operations.
Lecture 3: Isomorphism of binary structures. Groups.
Lecture 4: Basic properties of groups. Semigroups.
Lecture 5: Subgroups. Order of an element in a group. Cyclic groups.
Lecture 6: Cyclic groups (continued). Cayley graphs. Permutations.
Lecture 7: Cycle decomposition. Order and sign of a permutation.
Lecture 8: Sign of a permutation (continued). Classical definition of the determinant.
Cosets. Lagrange's theorem.
Part II: More advanced group theory
- Direct product of groups
- Factor groups
- Homomorphisms of groups
- Classification of abelian groups
- Group actions
Fraleigh/Brand: Chapters II and III
Lecture 9: Direct product of groups. Factor groups.
Lecture 10: Homomorphisms of groups.
Lecture 11: Classification of groups. Transformation groups.
Lecture 12: Review for Exam 1.
Lecture 13: Transformation groups (continued). Group actions.
Part III: Basic theory of rings and fields
- Rings and fields
- Integral domains
- Modular arithmetic
- Rings of polynomials
- Factorization of polynomials
Fraleigh/Brand: Chapters V and VI
Lecture 14: Rings and fields.
Lecture 15: Fields (continued). Advanced algebraic structures.
Lecture 16: Some examples of rings. Field of quotients.
- Fraleigh/Brand 22, 26, 32
Lecture 17: Modular arithmetic.
Lecture 18: Public key encryption. Rings of polynomials. Division of polynomials.
Lecture 19: Factorization of polynomials.
Lecture 20: Review for Exam 2.
Part IV: More advanced ring theory
- Ideals
- Factor rings
- Homomorphisms of rings
- Prime and maximal ideals
- Factorization in integral domains
- Euclidean algorithm
Fraleigh/Brand: Chapter VI
Lecture 21: Subrings and ideals. Factor rings.
Lecture 22: Homomorphisms of rings.
Lecture 23: Prime and maximal ideals. Ideals in polynomial rings.
Lecture 24: Factorization in integral domains. Principal ideal domains.
Lecture 25: Euclidean algorithm. Chinese remainder theorem.
Lecture 26: Review for the final exam.
- Fraleigh/Brand 0-14, 22-24, 26-28, 30-32