# Math 401

## Fall 2014

A first course in rings and groups out of Shifrin.

- Syllabus.
- Errata for the textbook.
- Homework 1 (due 9/2):
- Section 1.1: #4 (choose four parts), 7, 8a, 16 (optional).
- Section 1.2: #1 (choose three parts), 6, 7, and the following variation on #11: Suppose that
*k* is not a square. Show that there are no integers *m* and *n* such that *m*^{2} = *k n*^{2}. Hint: consider the prime factorization of *m* and *n*. Maybe warm up with *k* = 2 and *k* = a prime.
- Solutions: pdf, tex.

- Homework 2 (due 9/11):
- Section 1.3: #14, 15, 16 (optional), 20 (choose three parts), 21 (choose three parts), 25, 38c. Read through all the problems in the section.
- Solutions: pdf, tex.

- Homework 3 (due 9/18):
- Section 1.4: #3, 8, 11, 19bd, 20, and the following variation on #7: Suppose that
*R* is any ring, *c, x, y ∈ R*, and *c* is neither zero nor a zero-divisor. Prove that if *cx = cy* then *x = y*.
- Section 2.1: #8, 15, and show that the sets
*R* appearing in #12 and 13 are indeed rings, i.e. they contain 1 and are closed under addition, subtraction, and multiplication.
- Section 2.2: #5, 6ce.
- Solutions: pdf, zip.

- Handout on Writing by Jack Lee.
- Homework 4 (due 9/25):
- Section 2.3: #6, 9 (choose three parts), 10, 17.
- Section 2.4: #1 (choose two parts), 3, 6 (but don’t worry about Δ and Proposition 4.4), 7 (choose either part), 10, 11 (optional).
- Solutions: pdf, tex.

- What is the smallest prime?
- Homework 5 (due 10/2):
- Section 3.1: #1 (choose one of c,d,e), 6, 10 (all parts), 13, 14, 19, 20 (choose one of b,c).
- Solutions: pdf, tex.

- Homework 6 (due 10/16):
- Section 3.2: #6 (choose two parts), 7, 10, 14.
- Section 3.3: #2 (choose three parts), 3 (choose one part), 7, 10.
- Solutions: pdf, tex.

- Midterm 1, Solutions, tex.
- Homework 7 (due 10/23):
- Section 4.1: #4 (choose 3 parts), 10, 12, 15 (all parts), 16, 17a, and the following variation on 17b: In
**Z**, let *I* = (4) and *J* = (10). Identify *I* + *J* and *I* ∩ *J*.
- Beware that there are some printing errors in #16 – see the errata linked above.
- Optional: In class we saw that
**Z**_{2}[*x*]/(*x*^{2}+*x*+1) is a field with 4 elements. Similarly one can show that **Z**_{2}[*y*]/(*y*^{3}+*y*+1) and **Z**_{2}[*z*]/(*z*^{3}+*z*^{2}+1) are fields with 8 elements. Show that they are isomorphic.
- Solutions: pdf, tex.

- Homework 8 (due 10/30):
- Section 4.2: #1, 5, 11abc, 13, 22.
- Section 4.3: #1 (choose one part), 2 (choose three parts), 10 (choose two parts), 15.
- Solutions: pdf, tex.

- Homework 9 (due 11/6):
- Section 5.1: #3, 4, 11 (just give the answers, no proofs), 20.
- Section 5.2: #4 (he's referring to (1), (2), and (3) on page 159), 5.

- Midterm 2, Solutions, tex.
- Homework 10 (due 11/13):
- Make a tetrahedron and a cube out of paper. Determine the size of their symmetry groups.
- Solution: For the tetrahedron the answer is 24, or 12 if you take only rotations. For the cube it is 48, or 24 if you take only rotations.

- Homework 11 (due 11/20):
- Section 6.1: #1, 4, 5, 17, 18 (choose four parts).
- Section 6.2: #6b, 8, 12, and the following variation on #7: show that GL
_{2}(**Z**_{2}) ≅ S_{3}.
- Solutions: pdf, tex.

- Homework 12 (due 12/2):
- Section 6.3: #4, 5, 6, 12, 15, 23, 25, 31 (optional), 33.
- Solutions: pdf, tex.

- Homework 13 (not due):
- Section 6.4: #1, 5.
- Section 7.1: #3, 10. For #10, use the presentation Sq = { 1,
*r*, *r*^{2}, *r*^{3}, *s*, *sr*, *sr*^{2}, *sr*^{3} } with the relation *r*^{n}s = *sr*^{4-n}.

To say that a group acts on itself by conjugation means that we define *g⋅h = ghg*^{-1}.
- Section 7.2: #1.
- Solutions: pdf, tex.

- Final Exam, Solutions, tex.