Spring 2020

Instructor: Paul Hacking, LGRT 1235H, hacking@math.umass.edu

Meetings:

Classes: Tuesdays and Thursdays, 1:00PM--2:15PM in LGRT 145.

Office hours: Tuesdays and Wednesdays, 3:00PM--4:00PM in LGRT 1235H.

Course text: Abstract Algebra, by D. Dummit and R. Foote, 3rd ed., Wiley 2004. googlebooks.

Other useful references:

Introduction to commutative algebra by M. Atiyah and I. MacDonald. googlebooks.

Rings, modules, and linear algebra by B. Hartley and T. Hawkes. googlebooks.

Prerequisites: Math 611.

Homework:

Homeworks will be assigned every 1--2 weeks and posted on this page.

HW1. Due Thursday 2/13/20. Solutions.

HW2. Due Thursday 2/27/20. Solutions.

HW3. (will not be graded). Solutions.

Classlog here.

Exams:

There will be one midterm exam and one final exam.

The midterm exam will be held on Wednesday 3/11/20, 7:00PM--9:00PM, in LGRT 145.

The final exam will be a take home exam distributed on Tuesday 4/28/20 and due on Tuesday 5/5/20.

The algebra sequence 611--612 is also assessed via the algebra qualifying exam. Syllabus.

Your course grade will be computed as follows: Homework 30%, Midterm 30%, Final 40%.

Overview of course:

Here is the syllabus for 612. Roughly speaking it corresponds to Chapters 10--15 of Dummit and Foote. The full syllabus for the Graduate algebra sequence 611--612 is here.

(1) Modules (continued from 611).

Exact sequences. Hom. Bilinear forms. Tensor product of vector spaces, Abelian groups, and R-modules. Right-exactness of the tensor product. Restriction and extension of scalars. Symmetric and exterior algebras.

(2) Field Theory and Galois Theory.

Algebraic extensions. Finite extensions. Degree. Minimal polynomial. Adjoining roots of polynomials. Splitting field. Algebraic closure. Separable extensions. Theorem of the primitive element. Galois extensions. Fundamental Theorem of Galois Theory. Finite fields and their Galois groups. Frobenius endomorphism. Cyclotomic polynomial. Cyclotomic fields and their Galois groups. Cyclic extensions. Lagrange resolvents. Solvable extensions. Solving polynomial equations in radicals. Norm and trace. Transcendence degree.

(3) Commutative algebra.

Chain conditions. Noetherian rings. Hilbert's Basis Theorem. Prime and maximal ideals. Localization of rings and modules. Exactness of localization. Local rings. Nakayama's Lemma. Integral extensions. Noether's Normalization Lemma. Integral closure. Nullstellensatz. Closed affine algebraic sets.