Fall 2021
- MATH 415-501: Modern Algebra I
Time and venue: MWF 1:50–2:40 p.m., BLOC 160
Office hours (BLOC 301b):
- MWF 1:00–1:45 p.m.
- by appointment
Office hours (ZOOM meeting):
- Tuesday 5:00–6:00 p.m.
- by appointment
Additional office hours:
- Friday, December 10, 1:00–3:00 p.m. (ZOOM meeting)
- Monday, December 13, 1:00–3:00 p.m. (BLOC 301b)
- Tuesday, December 14, 1:00–3:00 p.m. (BLOC 301b)
- by appointment
Exam 1: Wednesday, October 13 (Sample problems)
Exam 2: Wednesday, November 10 (Sample problems)
Final exam: Tuesday, December 14, 3:30–5:30 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.
Lecture 2: Cardinality of a set.
Lecture 3: Binary operations.
Lecture 4: Isomorphism of binary structures. Definition of a group.
Lecture 5: Examples and properties of groups.
Lecture 6: Semigroups.
Lecture 7: Subgroups. Order of an element in a group.
Lecture 8: Cyclic groups. Cayley graphs.
Lecture 9: Cayley graphs (continued). Permutations.
Lecture 10: Cycle decomposition. Order of a permutation.
Lecture 11: Sign of a permutation. Classical definition of the determinant.
Lecture 12: 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 13: Direct product of groups. Factor groups.
Lecture 14: Factor groups (continued). Homomorphisms of groups.
Lecture 15: Isomorphisms of groups.
Lecture 16: Classification of groups.
Lecture 17: Transformation groups.
Lecture 18: Group actions.
Lecture 19: 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/Brand: Chapters V and VI
Lecture 20: Rings.
Lecture 21: Rings and fields.
Lecture 22: Advanced algebraic structures.
Lecture 23: Some examples of rings.
Lecture 24: Quaternions. Field of quotients.
Lecture 25: Modular arithmetic.
Lecture 26: Modular arithmetic (continued). RSA encryption.
Lecture 27: Rings of polynomials. Division of polynomials.
Lecture 28: Factorization of polynomials.
Lecture 29: Factorization of polynomials (continued).
Lecture 30: Review for Exam 2.
Part IV: More advanced ring theory
- Ideals
- Factor rings
- Homomorphisms of rings
- Prime and maximal ideals
- Factorization in general rings
Fraleigh/Brand: Chapter VI
Lecture 31: Subrings and ideals.
Lecture 32: Factor rings. Homomorphisms of rings.
Lecture 33: Homomorphisms of rings (continued).
Lecture 34: Isomorphism of rings. Prime and maximal ideals.
Lecture 35: Ideals in polynomial rings.
Lecture 36: Factorization in integral domains.
Lecture 37: Principal ideal domains. Euclidean algorithm.
Lecture 38: Chinese remainder theorem.
Lecture 39: Review for the final exam.
- Fraleigh/Brand 0-14, 22-24, 26-28, 30-32
Lecture 40: Review for the final exam (continued).
- Fraleigh/Brand 0-14, 22-24, 26-28, 30-32