# Math 407     Abstract Algebra     Fall 2014

## Instructor: Sunil Chebolu

• Office: Stevenson 337
• Email: szchebol@ilstu.eduz (with the two "z"'s removed)
• Office Hours: Open door and also by appointment
• Class meeting time:
• Tuesdays and Thursdays in STV 126
• Text: Galois Theory (3rd Edition) by Ian Stewart.

Announcements:

• Welcome to MAT 407! You first job is to get a physical copy of the text book either from an online bookstore or from a library. Make sure you get the 3rd edition. I heard that the earlier editions of this book are significantly different from the 3rd edition. A soft copy is available below.

Projects: I will collect here some ideas for inclass projects. These will be assigned to students on a first-come-first-serve basis.

• Lagrange's method for solving polynomial equations of degree less than 5 which is based on some general principle about the symmetric functions of the roots.
• An in-depth analysis of cubic and quartic equations.
• Gauss's work on the 17-gon
• Hilbert's theorem 90.

Extra Credit Problems: There are no due dates for these problems. You can hand them in whenever you want.

1. 3.16 (you may use Sage to find the answer here)
2. 6.11

## Homework

• Nov 13 (due Nov 25th)
1. Read Chapter 15.

• Nov 8 (due Nov. 13th)
1. Read Chapters 13 and 14

• Oct 24 (due Nov. 6th)
1. Read Chapter 12.
2. Turn in problems 11.1, 11.6, and do parts (a) and (e) for 11.2, 11.3, and 11.4
3. Start working on your final projects.

• Oct 15 (due Oct. 23rd)
1. Read Chapter 11
2. Work on the take-home midterm

• Oct 4 (due Oct. 14th)
1. Do all problems. Turn in 9.1, 9.2, 9.6, and 9.7.
2. Read Chapter 10.

• Sep 25 (due Oct. 2nd)
1. Prepare for your presentations (on Chapter 7) to given on October 7th.
2. Read Chapter 8 (Section 8.1 -- 8.6). Skip the rest of this chapter for now.
3. Turn in problems 8.1, 8.12 (parts a--d)
4. Turn in the proofs of the inclusions in (8.2).
5. Read Chapter 9.

• Sep 18 (due Sep. 25th)
1. Complete reading all chapters up to 6.
2. Do all problems of Chapter 6 except 6.5, 6.11, 6.15, and 6.16.
3. Turn in the following problems: 6.1 (c),(d), 6.3, 6.6, 6.8, and 6.9

• Sep 9 (due Sep. 18th)
1. Read Chapter 6.
2. Do all problems of Chapter 5.
3. TBA

• Sep 2 (due Sep. 9th)
1. I made one mathematical mistake in my lecture today. Read your notes carefully and find it.
2. Try to fix that mistake if you can.
3. Read Chapter 5
4. Sage: Given a non-zero element in the field $$K = \mathbb{Q}(2^{1/3})$$ find an explicit formula for its inverse.
5. Do the following problems: 4.2, 4.5 (this involves some basic facts about infinite cardinals), 4.10
6. Turn in the following problems: 4.3, 4.4, and 4.8

• Aug 28 (due Sep. 2nd)
1. Read Chapter 4.
2. Sage: Input a polynomial $$f(x)$$ in $$\mathbb{Z}[x]$$ and get all primes $$p < 1000$$ for which $$f(x) \mod p$$ is irreducible. (What happens when you take $$f(x) = x^4+1?$$)

• Aug 27 (due Sep. 2nd)
1. 3.1 a, 3.2 (find hcf for the polynomials in 3.1 (a)), 3.4 (use theory and then check your answers using sage), 3.5 a, 3.17 (give proper justification) [Turn these for grading]
2. Introductory Programming Tutorial 2 in Sage Watch this video

• Aug 22: (due Aug 26th)
1. Skim through Chapter 3 before coming to class on Tuesday.
2. Start playing with Sage: Input a 3rd degree polynomial and ask Sage to find its roots, discriminant, and plot the polynomial.
3. Introductory Programming Tutorial in Sage Watch this video

• Aug 19: (due Aug 26th)
1. Read the above-mentioned paper by Jack Lee.
2. Read Historical Introduction
3. Read Chapter 1 and do exercise 1.2, 1.5.
4. Read Chapter 2 (skip the proof of FTA. Later in this course we will have a chance to see a different proof of FTA.) and do exercises 2.1 and 2.2.
5. Introduction to Sage Watch this video on introduction to sage and set up a sage notebook and cloud accounts.