# Math 210B: Modern Algebra

This is the second course in a three-part sequence. From the course guide: ``Galois theory. Ideal theory, introduction to algebraic geometry and algebraic number theory.'' The course will assume that you are very comfortable with the material of Math 210A. We'll cover much of the material in chapters V, VI, VII, IX, and X of Lang's Algebra: field theory (including Galois theory) and some selected topics in commutative algebra.

Lectures: Tuesdays and Thursdays 1:15-2:30 in 381-T.

Office hours: Tuesdays and Thursdays 2:30-4:00 in 383-M (third floor of the math building). I will have bonus office hours before the midterm and final.

Textbook: Lang's Algebra (revised third edition). Other references: Dummit and Foote, and Emil Artin's little book on Galois theory. Artin's book is now available for free.

• 7-8 weekly problem sets 40%.
• In-class mid-term 20%. Here is a practice midterm.
• Final exam 40%. Here is a practice final. The final exam will take place on Monday, March 19, 3:30-6:30 pm, in our usual classroom, 381-T.

Course assistant: Jarod Alper, jarod@you-know-where.edu, office 380-J.

Problem sets: They will be mostly due on Fridays at noon, at Jarod Alper's door (380-J). There will be an envelope there. Lates will not be allowed, but the lowest score will be dropped. Problem sets in ps and pdf formats are below. (Please let me know if you have trouble with the pdf version, or if you want the dvi version.)

• Problem set 1 out (ps, pdf, due Mon., Jan. 29). Here is Brian Krummel's answer to problem 6.
• Problem set 2 out (ps, pdf, due Fri., Feb. 2).
• Problem set 3 out (ps, pdf, due Fri., Feb. 9).
• Problem set 4 out (ps, pdf, due Fri., Feb. 23).
• Problem set 5 out (ps, pdf, due Fri., Mar. 2).
• Problem set 6 out (ps, pdf, due Fri., Mar. 9).
• Problem set 7 out (ps, pdf, due Fri., Mar. 16).

The course so far:

• Class 1 (Tu Jan. 9): motivation: 3 impossibilities of classical greece. Introduction to fields.
• Class 2 (Th Jan. 11): field extensions and how to think about them, degree, algebraic.
• Class 3 (Tu Jan. 16): splitting fields, adjoining algebraic elements, algebraic extensions, quadratic extensions in characteristic not 2, classification of finite fields.
• Th Jan. 18 class cancelled due to illness.
• Class 4 (Tu Jan. 23): finite fields continued; separable elements of an extension, separable extensions, in terms of maps to algebraic closure.
• Class 5 (Th Jan. 25): separability continued, automorphisms of an extension, Galois extensions and Galois groups, examples.
• Class 6 (Tu Jan. 30): theorem of symmetric functions, introduction to group characters.
• Class 7 (Th Feb. 1): transcendence bases, group characters, G-invariants of a field are index |G|.
• Class 8 (Tu Feb. 6, taught by Greg Brumfiel): fundamental theorem of Galois theory; introduction to cyclotomic polynomials.
• Class 9 (Th Feb. 8): Galois theory in action.
• Clas 10 (Tu Feb. 13): Solvability questions.
• In-class mid-term (Th Feb. 15).
• Class 11 (Tu Feb. 20): more solvability; composite extensions and simple extensions.
• Class 12 (Th Feb. 22, taught by Dragos Oprea): cyclotomic and abelian extensions; beginning of commutative ring theory, Noetherian rings, Hilbert basis theorem.
• Class 13 (Tu Feb. 27): affine n-space, the maps between subsets of n-space and ideals, morphisms of algebraic setes (in correspondence with homorphisms of rings).
• Class 14 (Th Mar. 1): radical ideals, the Zariski topology, existence and uniqueness of decomposition into irreducibles.
• Class 15 (Tu Mar. 6): primary ideals, primary decomposition.
• Class 16 (Th Mar. 8): Spec R and its Zariski topology, integral extensions.
• Class 17 (Tu Mar. 13): integral extensions, Noether normalization, and Hilbert's Nullstellensatz.
• Class 18 (Th Mar. 15): Hilbert's Nullstellensatz, three facts about integral extensions, localization.