Commutative Algebra and Galois Theory
Prof: Andrei Caldararu
Grader: Evan Dummit
Caldararu's office hours: Monday 1:30pm.
Grader's office hours: Wednesday 2:15pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.
This course, the second semester of the introductory graduate sequence in algebra, will cover the basic aspects of commutative ring theory and Galois theory. The textbook we'll use for the Commutative Algebra portion will be Atiyah-Macdonald "Commutative Algebra". For Galois Theory I plan on using Emil Artin's notes which are available here, but I may change my mind before we start on it.
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course.
Commutative rings and homomorphisms, integral domains, fields. Ideals, prime and maximal. Existence of maximal ideals. Local rings. Some geometric pictures (Spec and Specm). Nilradical. The nilradical is the intersection of all primes. Relatively prime ideals, Chinese remainder theorem. Extension and contraction, pictures of what happens for the inclusion Z -> Z[i]
Modules, module homomorphisms. Module Hom, covariance and contravariance. Finitely generated modules, Nakayama's lemma and variants. Short exact sequences. Tensor product and exactness properties.
Below you will find a repository of homework problems.
HOMEWORK 1 (due Feb 4)
Atiyah-Macdonald, page 10: 1, 2, 6, 10, 12, 15, 16, 17, 18, 21
HOMEWORK 2 (due Feb 15)
Atiyah-Macdonald, page 10: 19, 22, 26, 27, 28; page 31: 2, 3, 4, 8, 9