Math 172
Abstract Algebra II

Professor Francis Su
x73616, [my last name]@math.hmc.edu
Course meets: MWF 10am, Beckman B134
Office Hours: Mondays 11-12:15pm, and 2:30-3:00pm, or by appointment.

Grader: Sara Gussin, [firstname_lastname]@hmc.edu

Course Content: The main topics of this course will be field extensions and Galois Theory, and additional topics as time permits.

Text: Dummit and Foote, Abstract Algebra. Doing the reading will be essential for success in this course. Also as a second-semester course, I will expect a certain level of mathematical maturity. This means that sometimes I will not prove simple statements in class; you should get in the habit of working out some details for yourself, and doing the reading.

Prerequisite: Algebra I (Math 171).

Homeworks: Homeworks, assigned weekly, turn in by Tuesdays 2pm in the bin outside my office door. Homeworks will be announced on the course webpage: http://www.math.hmc.edu/~su/math172/

Grading policy: A midterm (worth 25%), a final (worth 35%) and a homeworks (40%) with the lowest homework score dropped.

Honor Code: All are expected to abide by the HMC honor code. Cooperation is ENCOURAGED in this class, but write up all solutions individually and be sure to credit any collaborators.

HW Assignments
HW #1 (due 1/29) REVIEW 7.1-7.4 and READ 7.5
(R means READ but you do not have to do the problem.)

DO 7.3 ( 1, 10, R19, R26, 28 ) 7.4 ( 10 ) and Problem A:
A. (part i) Let R be an integral domain. As you conjectured in class, prove that the units in R[x] are precisely the constant polynomials p(x)=u where u is a unit in R.
(part ii) On the other hand, show that p(x)=1+2x is a unit in R[x], where R=Z/4Z. (Z= integers.)

HW #2 (due 2/5) SKIM CHAPTER 8, READ 9.1-9.3
DO 7.4 ( 19 ) 7.5 ( 3 ) 8.2 ( 2, 3 ) 9.1 ( 4 )

[You will probably find 7.5.3 to be the most involved problem; so you may wish to do it last. For several of the others, I encourage you to think about modding out a ring by an ideal.]

HW #3 (due 2/12) READ 9.4-9.5
DO 9.1 ( R2, R6, 7 ) 9.2 ( 1, 2, 3, R4 ) 9.3 ( 3 )
HW #4 (due 2/19) READ 13.1-13.2
DO 9.2 (R6) 9.3 ( 4 ) 9.4 ( 1bcd, 2bc, R5, R6ac, 7, 11, R20 ) 13.1 ( 1 )
HW #5 (due 2/26) READ 13.4
DO 13.1 ( 3, 5 ) 13.2 ( R1, 3, 4, 12, 14 )
HW #6 (due 3/4) READ 13.4-13.6
DO 13.3 ( 1, R2 ) 13.4 ( 2, 3, 4 ) and these problems:

Problem B. Find a real number u such that Q(sqrt[3],sqrt[5])=Q(u).

Problem C. Suppose K is an extension of F, and phi: K -> K is an isomorphism that leaves every element of F fixed.
Show that any polynomial in F[x] that has a root r in K also has phi(r) as a root.

MIDTERM (due Friday 3/14) and no HW over Spring Break.
HW #7 (due 4/1) READ 14.1-14.2, 14.6
DO 14.1 ( 7 ) 14.2 ( 3, 13, 14 )
HW #8 (due 4/8) READ 14.6
DO 14.6 ( 22, R23, 37, 38, 39, 40 )
HW #9 (due 4/15) READ 14.7
DO 14.6 ( 2bcd, 4, R29-32, 46 )
In problem 46, E and F are unspecified fields (that you get to specify).
Problem 14.7 ( 4 ) has been removed from the HW.
HW #10 (due 4/22) SKIM 9.6, 15.1. The main ideas are here, but I'm supplementing what's here with my lectures.
DO homework on this handout. The TeX source can be found here.
HW #11 (due 4/29) You will want to get the class notes from someone if you've missed any lectures, as we have been departing from what's in the text to follow a more motivated treatment of Groebner bases via monomial ideals.

DO homework on this handout. The TeX source can be found here.

This is your final homework!