# Math 121: Modern Algebra II

This is the second course in a two-part sequence. From the course
guide: ``Continuation of 120. Fields of fractions. Solvable
and simple groups. Elements of field theory and Galois theory.
Prerequisite: 120.''
**Lectures:** Tuesdays and Thursdays 9:30-10:45 in 380-D.

**Course assistant:** Jason Lo
jasonlo@math.you-know-where.edu,
office 380-U1.

**Office hours:** My office hours will
be after class, Tuesdays and Thursdays 10:45-12:15, in my office
383-M (third floor of the math building).
Jason's office hours are Mondays and Wednesdays 3:30-5 pm.
There will be bonus office hours before the midterm and final.

** Text:** Dummit and Foote's *Absract Algebra* (second
edition), the same book as in math 120. It's possible that
the book in the bookstore (and that you own) is the third edition;
that should make no difference, but please warn me if there
are any differences.

**Grading scheme:**

7-8 weekly problem sets 40%.
In-class mid-term 20%. (Possible date: Thurs. February 1, so
grades will be back in time for the drop deadline of Sun. February 4,
or else as late as Thurs. February 15.) Here is a practice midterm. Here is the midterm.
Final exam 40%.
Here is a practice final.
The final exam will take place on Wednesday, March 21, 12:15-3:15 pm,
in our usual classroom, 380-D.
**Problem sets:** In general, they will be due on Friday at noon in Jason
Lo's mailbox on the main floor of the math department.
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.) Sketches of solutions will be posted
on Jason Lo's homepage.
Lates will not be allowed, but the lowest score will be dropped.

Problem set 1 (ps, pdf,
updated Jan. 25) due Mon., January 29 at 6 pm.
Problem set 2 (ps, pdf),
due Fri., February 2.
Problem set 3 (ps, pdf),
due Fri., February 16.
Problem set 4 (ps, pdf),
due Fri., February 23. (Caution: the page numbers refer to the
2nd edition,. John Le says add 20 pages to each reference for edition 3. The first
three problems are from 14.2, and the last is from 14.6.)
Problem set 5 (ps, pdf),
due Fri., March 2.
Problem set 6 (ps, pdf),
due Fri., March 9.
Problem set 7 (ps, pdf),
due Fri., March 16.
**The course so far:**

Class 1 (Tu Jan. 9): motivation: three (or four)
impossibilities from classical Greece.
Class 2 (Th Jan. 11): fields, extensions, charactetristic, prime subfield, degree, algebraic element.
Class 3 (Tu Jan. 16): how to think about adjoining an algebraic element.
Class Th Jan. 18 cancelled due to illness.
Class 4 (Tu Jan. 23): splitting fields, properties of field extensions.
Class 5 (Th Jan. 25): examples and applications: quadratic extensions,
finite fields.
Class 6 (Tu Jan. 30): finite fields of each possible order exist; inclusion
of one finite field in another; algebraic closures exist and are unique
up to non-unique isomorphism.
Class 7 (Th Feb. 1): separable polynomials, perfect fields, separable
field extensions, map-separability.
Class 8 (Tu Feb. 6, taught by Matt Young): separable parts of extensions, introduction to Galois theory.
Th Feb. 8: In-class midterm.
Class 9 (Tu Feb. 13): the number of automorphisms of an extension
is at most the degree; Galois := equality; examples.
Class 10 (Th Feb. 15, taught by Jun Li): the symmetric function example.
Introduction to characters.
Class 11 (Tu Feb. 20): statement of the fundamental theorem of galois
theory; examples.
Class 12 (Th Feb. 22, taught by Jason Lo): proof of the fundamental
theorem of galois theory.
Class 13 (Tu Feb. 27): cyclic extensions, ruler and compasses take two, insolubility of the quintic. **Here is a great visual discription of the
construction of the pentagon, by TokyoJunkie on Wikipedia.**
Class 14 (Th Mar. 1): solving the cubic and quartic; cyclic extensions
are obtained by taking nth roots (if the base field has roots of unity).
Class 15 (Tu Mar. 6): using Galois theory to figure out how to
solve the cubic by hand.
Class 16 (Th Mar. 8): solving the quartic by hand, and Galois
groups of quartics.
Class 17 (Tu Mar. 13): symmetric function theorem (redux), composite extensions.
Class 18 (Th Mar. 15): simple extensions and the primitive element theorem, conclusion of course.

To my home page.

Ravi Vakil

Department of Mathematics Rm. 383-M

Stanford University

Stanford, CA

Phone: 650-723-7850 (but e-mail is better)

Fax: 650-725-4066

vakil@math.you-know-where.edu