Course Description
More advanced abstract algebraic structures and concepts. Further study of group theory, including finite abelian groups and the Sylow theorems. Field extensions. Further applications of group theory, including Galois theory and geometric constructions. Arithmetic in integral domains. Additional topics such as public key cryptography or algebraic coding theory, as time permits.
Additional Requirements for Graduate Students:
Extra problems on weekly homework.
Athena Title
Modern Algebra II
Prerequisite
MATH 4000/6000
Semester Course Offered
Offered spring
Grading System
A - F (Traditional)
Student learning Outcomes
- Students will be able to classify finite abelian groups, and study general groups of small order.
- Students will be able to state and apply the Sylow theorems concerning subgroups of certain orders in a given finite group.
- Students will relate groups to polynomial rings through Galois theory, and use this to explain why there cannot be an algebraic formula for the roots of a general polynomial of degree 5 and higher.
- Students will perform calculations in and prove results about specific integral domains such as rings of quadratic integers, as well as general integral domains of various types such as PIDs.
Topical Outline
- Topics in Group Theory, including finite abelian groups and the Sylow theorems
- Field extensions
- Galois theory
- Geometric constructions
- Arithmetic in integral domains (including PIDs, quadratic integers)
- Additional topics such as public key cryptography or algebraic coding theory, as time permits
