Daniel Erman (Spring 2018):
(Caldararu, 2014) Math 742 is the second half of the basic first-year graduate course in abstract algebra. It stresses the commutative aspects of the subject and consists of two main parts. The first part covers the basics of commutative algebra, including the Hilbert basis theorem and Nullstellensatz, and basic geometric notions including dimension and primary decomposition of ideals. The second part concerns field theory and includes Galois theory, constructability of geometric objects, solvability of polynomials, algebraic number fields. We shall also discuss a little bit of non-commutative algebra, including Wedderburn theory and the basics of Morita theory. Textbook: For the first half of the course, Atiyah-Macdonald "Commutative Algebra". For the second half, "Fields and Galois Theory" by J.S. Milne, available freely online.