Fields and Galois Theory. Tentative Syllabus

The lowest homework grade will be dropped. No late homework will be accepted.

- WEEK 1 (Jan 5-Jan 9): Introduction: an overview of ruler and compass construction, solvability by radicals.

Review of rings, fields, polynomials.

Homework 1, due in class on Monday January 19th.

- WEEK 2 (Jan 12-Jan 16): Quotient rings, algebras. Algebraic (and transcendental) elements, minimal polynomial, degree of an algebraic element (worksheet 1).

Homework 2, due in class on Monday January 26th.

- WEEK 3 (Jan 19-Jan 23): Stem field, algebraic closure. The ring Z[X] (worksheet 2).

- WEEK 4 (Jan 26- Jan 30): Splitting field of a polynomial. Finite fields. Homework 3 and a few extra problems Homework 3 is due in class on Friday February 13th.

- WEEK 5 (Feb 2nd-Feb 6): Galois correspondence for perfect fields. Review 1

- WEEK 6 (Feb 9-Feb 13): Galois group of a polynomial.

Midterm Exam Solution

Homework 4 due on Wednesday February 25th.

MIDTERM BREAK

- WEEK 7 (Feb 23-Feb 27): Cyclotomic fields (see also Homework 4). Galois group of an algebraic closure of a finite field: worksheet 3.

- WEEK 8 (March 2nd-March 6): More on worksheet 3. Solvable groups. Solvability by radicals. Worksheet 4.
- WEEK 9 (March 9-March 13): Algebraic integers. Decomposition subgroups. Homework 5 due on March 18th.

- WEEK 10 (March 16-March 20): Review 2 Midterm 2 Solution

- WEEK 11 (March 23-March 27): Algebraic integers in a quadratic extension. Worksheet 5

Homework 6 due on April 8: Problem 3 of Midterm 2, Problems 4 and 5 of Worksheet 5.

- WEEK 12 (March 30-April 3): Review problems

- WEEK 13 (April 6-April 10):

FINAL EXAM: April 20th, 12pm-2:30pm MATH 204 NOTE THIS IS NOT OUR USUAL ROOM