## ANNOUNCEMENTS

Registration for Winter term opens Thursday, November 17, 2016.

This page was last updated: .

## COURSE DESCRIPTION

The second term of Math 120 will be devoted primarily to Field and Galois Theory. Time permitting, topics should include:

**Field extensions:** Algebraic closure. Splitting fields and normal extensions. Separable and Inseparable extensions and degree. Primitive element theorem. Linear independence of characters. Traces and norms. Normal basis theorem.

**Galois theory**: Galois extensions. Absolute Galois group. Galois theory for infinite extensions.

**Examples:** Finite fields. Cyclotomic fields.

**Galois Cohomology:** Galois Cohomology. Hilbert's 90 theorem.

**Kummer Theory:** Cyclic extensions. Solvable extensions. Kummer Theory.

**Transcendental Extensions:** Transcendental extensions. Transcendence bases theorem.

**Ring extensions**: Integral extensions. Noether Normalization Theorem. Integral closure. Integral Galois extensions. Going-up theorem. Decomposition groups. Extension of homomorphisms. Valuation rings.

**Complements:** Introduction to local fields. Dedekind domains. DVR's.

## SCHEDULE

MWF 09:00 - 09:55, 257 Sloan.

## INSTRUCTORS

Eric Rains

PROFESSOR OF MATHEMATICS

Sloan 286

626-395-4322

## POLICIES

Grading: Based on homework; no exams.

Homework: There will be four homeworks due roughly every two weeks. They will definitely be longer than a one-week homework. The first three homeworks will have roughly equal weight and the last homework, which will have a three-week time period, will be longer and be worth somewhat more. Until one week before the homework is due, you may not collaborate but you may use any books. For the last week you may collaborate with others in the class or ask leading questions of the TA or Professor. The spirit of this is that you should be making significant effort to do the homework yourself before speaking to others (so if you leave the homework for the last few days, you really shouldn't collaborate until a day after you seriously start). Also, this should be primarily your own efforts. So it is contrary to the spirit (but not technically the
letter) of the collaboration policy for two or more students to sit down and do the homework as a joint project. Included in "use any books" is googling on the Internet. However, posting the homework on a chat site is definitely an example of collaboration NOT with others in the class and is therefore not allowed.
Late homework policy: If you get a note from the Dean's Office (personal problems) or infirmary (medical problems) requesting a postponement, it will be honored. Otherwise, late homework will not be accepted EXCEPT ONE time (per term) you may email me PRIOR to the due time requesting an extension of up to one week (for the last HW up to two days) and it will granted.

## TEXTBOOKS

Algebra, Serge Lang, Springer, ISBN: 978-0-387-95385-4