Spring 2021
- MATH 415-500: Modern Algebra I
Time and venue: TR 1:30–2:45 p.m., ZOOM meeting
Office hours (ZOOM meeting):
- MW 4:00–5:00 p.m.
- TR 12:15–1:15 p.m.
- by appointment
Office hours during the finals (ZOOM meeting):
- Friday, April 30, 1:00–3:00 p.m.
- Monday, May 3, 3:00–5:00 p.m.
- Tuesday, May 4, 3:00–5:00 p.m.
- Wednesday, May 5, 3:00–5:00 p.m.
- by appointment
Final exam: Thursday, May 6, 8:00–11:00 a.m. (Sample problems)
Course outline:
Part I: Basic group theory
- Binary operations
- Groups
- Subgroups, cyclic groups
- Groups of permutations
- Cosets, Lagrange's theorem
Fraleigh: Chapters I and II
Lecture 1: Preliminaries.
Lecture 2: Binary operations.
Lecture 3: Isomorphism of binary structures. Groups.
Lecture 4: Groups and semigroups. Subgroups.
Lecture 5: Generators of a group. Cyclic groups. Cayley graphs.
Lecture 6: Permutations. Cycle decomposition.
Lecture 7: Order and sign of a permutation.
Lecture 8: Definition of the determinant. Cosets. Lagrange's theorem.
Part II: More advanced group theory
- Direct product of groups
- Classification of abelian groups
- Homomorphisms of groups
- Factor groups
- Group actions
Fraleigh: Chapters II and III
Lecture 9: Direct product of groups. Factor groups.
Lecture 10: Homomorphisms of groups. Classification of groups.
Lecture 11: Classification of groups (continued). Groups of symmetries. Group action on a set.
Lecture 12: Review for Exam 1.
Part III: Basic theory of rings and fields
- Rings and fields
- Integral domains
- Modular arithmetic
- Rings of polynomials
- Factorization of polynomials
Fraleigh: Chapter IV
Lecture 13: Rings and fields.
Lecture 14: Follow-up on Exam 1. Advanced algebraic structures.
Lecture 15: Rings and fields (continued). Field of quotients.
Lecture 16: Modular arithmetic.
Lecture 17: Rings of polynomials. Division of polynomials.
Lecture 18: Factorization of polynomials over a field.
Lecture 19: Review for Exam 2.
Part IV: More advanced ring theory
- Ideals
- Homomorphisms of rings
- Factor rings
- Prime and maximal ideals
- Factorization in general rings
Fraleigh: Chapters IV and V
Lecture 20: Ideals and factor rings.
Lecture 21: Follow-up on Exam 2. Homomorphisms of rings.
Lecture 22: Homomorphisms of rings (continued). Prime and maximal ideals.
Lecture 23: Factorization in integral domains.
Lecture 24: Euclidean algorithm. Chinese remainder theorem.
Lecture 25: Review for the final exam.