Math 452 - Abstract Algebra II
Description
Math 452 is the second course in the honors algebra sequence, 451/452. The class will continue where 451 left off, starting with the study of rings. Then it will study either modules, fields, and Galois theory depending on time.
Topics covered
Topics usually covered in Math 452 include:
- Rings: definition and examples, ring homomorphisms and isomorphisms, ideals, polynomial rings, product and quotient rings, isomorphism and correspondence theorems
- Factorization in rings: prime vs. irreducible elements of a ring, Euclidean domains, principal ideal domains (PIDs), unique factorization domains (UFDs)
- Modules: basic definitions and examples, span and linear independence in a module, finitely generated vs. free modules, modules over PIDs, etc.
- Fields: fields and field extensions, algebraic vs. transcendental elements and extensions, degree of a field extension, finite fields, etc.
- Galois theory: fixed fields, Galois correspondence, and applications
Note that the exact list of topics may vary by instructor and semester.
From the above list, rings and factorization in them as well as fields are almost always covered, and typically one topic out of modules and Galois theory is covered.
Resources
Textbook
The usual textbook for the class Algebra (2nd edition) by Michael Artin.
Additional materials
Notes taken from a comparable course to Math 452 at MIT are available online here.
Other useful textbooks include Abstract Algebra by David Dummit and Richard Foote, a more advanced textbook typically used in graduate algebra courses but has similar topic coverage.
A course webpage from when the course was taught by Dima Sinapova in Spring 2025 may be found online at https://sites.math.rutgers.edu/~ds2005/Math%20452-S25.