# Abstract Algebra Dummit Foote Homework Solutions

## Math 332: Abstract Algebra, Spring 2015

MTWF 10-10:50am, Library 389Office Hours: TW 11-noon, θ 2:30-3:30pm, Library 313

Textbook: *Abstract Algebra*, 3rd edition, by Dummit and Foote.

syllabus.pdf / syllabus.tex

Compiled course notes: notes.pdf. Click on a day of the week to get notes for that day.

### Week 1: Jan. 26 - 30

- Monday: Review the syllabus. Groups: definitions and examples.
- Tuesday: Read pp.16-27. Dihedral groups.
- Wednesday: Symmetric groups. Matrix and quaternion groups.
- Friday: Read pp.29-39. Homomorphisms, isomorphisms, monomorphisms, epimorphisms, endomorphisms, and automorphisms.

#### Homework due Monday, Feb. 2

### Week 2: Feb. 2 - 6

- Monday: Read pp.41-52. Group actions.
- Tuesday: Read pp.54-64. Subgroups generated by subsets.
- Wednesday: Read pp.73-85. Quotients and cosets.
- Friday: Reread pp.73-85. More on quotients and cosets.

#### Homework due Monday, Feb. 9

### Week 3: Feb. 9-13

- Monday: Read pp.89-95. Cosets and Lagrange's theorem.
- Tuesday: Read pp.97-100. Isomorphism theorems.
- Wednesday: Read pp.101-105. Composition series and the Hölder program.
- Friday: Read pp.106-110. The alternating group.

#### Homework due Monday, Feb. 16

### Week 4: Feb. 16-20

- Monday: Read pp.112-121. Group actions redux (faithful actions, orbit decomposition).
- Tuesday: Read pp.122-129. The class equation.
- Wednesday: Conjugacy in
*S*._{n} - Friday: Read pp.133-137. Automorphisms.

#### Homework due Monday, Feb. 23

### Week 5: Feb. 23-27

- Monday: Read pp.139-142. The Sylow theorems.
- Tuesday: Read pp.142-146. More Sylow.
- Wednesday: Party tricks with small groups.
- Friday: Read pp.149-150. The simplicity of
*A*. Handout on other proofs of_{n}*A*'s simplicity by Keith Conrad._{n}

#### Homework due Monday, March 2

### Week 6: March 2-6

- Monday: Read pp.152-165. Direct products, categorical products.
- Tuesday: Read pp.169-173. Commutators and abelianization.
- Wednesday: Functors and adjunctions. Recognizing direct products.
- Friday: Test review.

#### Review exercises

### Week 7: March 9-13 (guest instructor: Dave Perkinson!)

- Monday: 120-minute take-home exam: exam1.pdf / exam1.tex. (DO NOT LaTeX your exam, please.)
- Tuesday: Read pp.175-184. Semi-direct products.
- Wednesday: Read pp.215-218. Free groups.
- Friday: Read pp.218-220. Presentations.

#### Homework due Monday, March 16

#### Math 332 concepts we have covered or [will cover] explicitly mentioned at Oberwolfach this week:

- groups, fields, [rings], cyclic groups, dihedral groups, symmetric groups,
*A*_{5}, direct products, semi-direct products, torsion subgroups, group actions, orbits, isotropy, categories, adjunctions, free groups, [ideals], [modules], [Galois extensions], [nilpotence], and much much more!

### Week 8: March 16-20

- Monday: Read pp.223-237. Rings.
- Tuesday: Read pp.239-247. Ring homomorphisms and quotients.
- Wednesday: Read pp.251-256. Ideals.
- Friday: Read pp.265-267. The Chinese Remainer Theorem.

#### Homework due Monday, March 30

### Spring Break: March 21-29

- Start contemplating your final project. Topic proposals are due via email by Friday, April 3.

### Week 9: March 30 - April 3

- Monday: Read pp.270-277. Euclidean domains.
- Tuesday: Read pp.279-282. Principal ideal domains.
- Wednesday: Read pp.283-294. Unique factorization domains.
- Friday: Sums of squares.

#### Homework due Monday, April 6

### Week 10: April 6-10

*Announcement*: Brunch for gender minorities in math and physics, April 11. RSVP to rhokaner@reed.edu or nuxolla@reed.edu by Monday, April 6.

- Monday: Read pp.295-298. Polynomial rings.
- Tuesday: Read pp.299-306. When is
*R*[*x*] a UFD? - Wednesday: Read pp.307-311. Irreducibility in polynomial rings.
- Friday: Read pp.313-315. Roots and factors.

#### Homework due Monday, April 13

### Week 11: April 13-17

- Monday: Read pp.337-343. Modules.
- Tuesday: Read pp.345-349. Quotients and hom groups.
- Wednesday: Read pp.351-355. Direct sums and free modules.
- Friday: Read pp.359-375. Tensor products.

#### Homework due Monday, April 20

### Week 12: April 20-24

- Monday: Tensor products, continued.
- Tuesday: Read pp.456-468. Modules over a PID.
- Wednesday: Proof the fundamental theorem (and donuts).
**Draft of final paper due in class.** - Friday: Read pp.472-488. Modules over polynomial rings.

#### Homework due Monday, April 27

### Week 13: April 27 - May 1

- Monday: The Cayley-Hamilton theorem.
- Tuesday: Read pp.4491-499. Jordan canonical form.
- Wednesday: Next steps, I. Course evals.
- Friday: Next steps, II.
**Final paper due at the start of class.**

#### Recommended exercises

- From Dummit & Foote: 12.2.3, 12.2.4, 12.2.7, 12.2.14, 12.2.20 (bonus), 12.3.5, 12.3.6, 12.3.10, 12.3.23, 12.3.31, 12.3.32
- These exercises will not be collected or graded, but you are encouraged to do them and discuss your solutions with the instructor.

### Review session: Wednesday, May 5, 1-3pm in Library 389.

### Three-hour take-home final: Monday, May 11

Pick up a paper copy of the final from the bin outside my office (Library 313) any time after 6am on Monday, May 11. Return the test before 10pm by sliding it under my office door. Three-hour time limit. Closed notes, closed book, closed internet, closed everything except for one 8.5" x 11" (two-sided) page of notes you've prepared in advance.

### Final papers

Send a pdf to ormsbyk@reed.edu if you'd like yours posted.

### The LaTeX document preparation system

Poor handwriting? Love escape characters? Too much free time? Try LaTeX!

Kyle M. Ormsby

## ANNOUNCEMENTS

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

This page was last updated: .

## COURSE DESCRIPTION

The second term of Math 5 will be devoted to the study of rings and modules; this includes some linear algebra. The material corresponds roughly to Chapters 7 through 12 in the text for the course, Abstract Algebra by Dummit and Foote. High points include the proof of unique factorization in principal ideal domains and the description of finitely generated modules over principal ideal domains.

## SCHEDULE

MWF 09:00 - 09:55, 151 Sloan.

## INSTRUCTORS

Elena Mantovan

PROFESSOR OF MATHEMATICS

EXECUTIVE OFFICER FOR MATHEMATICS

Sloan 262

626-395-4342

## OFFICE HOURS

Monday at 1:30pm in 262 Sloan Laboratory (EM)

Wednesday at noon in 382 Sloan Laboratory (XX)

## POLICIES

**Grading:** Roughly 40% Homework, 30% Midterm, and 30% Final.**Homework Policy:** Students may work together on homework as long as each student writes up their own homework in their own words.

In solving homework problems you will wish to quote theorems and lemmas from class and the text. Results from the text in sections not yet covered cannot be quoted unless you supply your own proof. The same holds for results from other texts not covered in class.

Many of the problems which will be assigned this year in Math 5, also were assigned in previous years. Do not consult old solution sets in working on this years homework.

At most one late homework set will be accepted, and except in unusual circumstances, only if arrangements have been made with me in advance. Other than in exceptional circumstances, homework will not be accepted more than a week after the deadline.

## TOPICS COVERED

Introduction to Rings, Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains, Polynomial Rings.

Introduction to Modules, Vector spaces, Modules over Principal Ideal Domains.

## TEXTBOOKS

D. Dummit and R. Foote, Abstract Algebra

## HOMEWORK

Due on Fridays at noon

Due Date | Homework | Solutions |
---|---|---|

01/13 | Set 1 | |

01/20 | Set 2 | |

01/27 | Set 3 | |

02/03 | Set 4 | |

02/10 at 10am | Midterm (see below) | |

02/17 | Set 5 | |

02/24 | Set 6 | |

03/03 | Set 7 | |

03/10 | Set 8 | |

03/16 at 10am | Final (see below) |

## EXAMS

Midterm exam was posted on Friday, February 3rd. An email was sent out with a link to the exam. If you have not received the email please contact PMA Undergraduate Student Affairs Administrator Meagan Heirwegh.

The midterm is due back by 10am on Friday February 10, 2017.

Final exam is posted here. An email was sent out with a password to the exam. If you do not received the email please contact PMA Undergraduate Student Affairs Administrator Meagan Heirwegh.

The final exam is due back by 10am on Thursday, March 16, 2017.

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