# Math 6130 (Fall 2015)

Algebra 1

## About the course

### From the course catalog

Studies group theory and ring theory. Department enforced prereq., MATH 3140. Instructor consent required for undergraduates. Requisites: Restricted to graduate students only.

### Unofficial description

This course is designed for incoming graduate students, and it is intended to prepare you for both the preliminary exams and your subsequent careers in mathematical research. There are some very specific mathematical goals, namely learning the basics of group theory and ring theory (essentially Chapters 1 through 9 of Dummit and Foote), but there are some more elusive ones too, such as developing your ability to read, write, and think about mathematics.

### Prerequisites

Very little prior *knowledge* of mathematics is required in this course. We will use some set theory, including functions, relations, and equivalence classes, and basic properties of the integers and congruence relative to a modulus. You can find a review of these topics in Chapter 0 ("Preliminaries") of Dummit and Foote, and I recommend studying it before the semester begins. If you are unsure of whether this class is appropriate for you, reviewing Chapter 0 will give you a feeling for the way things will be presented in class.

The other prerequisite is harder to pin down, but frequently gets called *mathematical maturity*. This means experience with proofwriting, abstract thinking, separating concepts from computations, communicating clearly, and probably means a lot of other things too. You develop it by thinking hard about math and explaining your thoughts to others, and you'll continue to develop it throughout this course and the course of your mathematical careers. However, it won't be possible to start from scratch in this course: a certain amount of mathematical maturity will be necessary from the beginning.

### Textbook

Dummit and Foote, *Abstract Algebra*, 3rd Edition

There are a lot of other great algebra textbooks. I encourage you to look at lots of different textbooks and find the one that suits you best. (Of course, it will be your responsibility to find the parts of your favorite textbook that are parallel to the assignments from Dummit and Foote.) Here is a partial list (you can make a much larger list with a bit of internet searching):

- Jacobson,
*Basic Algebra, I*, 2nd Edition - Lang,
*Algebra*, 3rd Edition - Hungerford,
*Algebra*, 8th Edition - Artin,
*Algebra*

### Grading

Homework: 1/3

Exam 1: 1/3

Exam 2: 1/3

## Office hours

I'll always have a few office hours every week, although the schedule may sometimes vary. Please make an appointment if you would like to speak privately, but feel free to drop in otherwise. Warning: I generally limit myself to 3 office hours per week. If those get used up or scheduled early in the week, I may not be available later.

## Assignments

You can't learn to speak a language by reading a textbook on grammar, and you can't learn to ride a bicycle by studying mechanics. In the same way, you can't learn algebra or any other mathematical subject just by reading a textbook and listening to lectures. You need to actively engage through the material, and one of the best ways to do that is with exercises.

Exercises will be posted each week, to be submitted *electronically* the following week. I will grade approximately four exercises each week from among those you submit (the actual number will depend on how much time the problems take to grade). Two of the problems I grade will be of your choice and the other two will be my choice. You should choose problems that are challenging but on which you have made meaningful progress.

I will grade your submissions based on the clarity of your writing, the correctness of your answers, and the appropriateness of the problems you have chosen.

Your grade on each assignment will be a single letter: A, B, C. These correspond roughly to `pass', `conditional pass', and `fail' on the prelims—not to the grades you will eventually receive in the course. I will supply comments, which I expect you to read and sometimes discuss with me (this is the whole point of my grading your exercises).

#### Guidelines for submitting assignments

- Assignments (after the first) must be typed using Latex and submitted electronically via D2L. You may submit the first electronically if you prefer.
**Do not submit homework by e-mail.** - Every assignment must include a section detailing references you have consulted. If you consult a reference and it contributes to what you write then you must cite it.
**Failure to do so is plagiarism!** - Indicate at the top of your submission the numbers of exactly two exercises you would like me to grade. If you don't do this, I will choose the two problems that seem easiest to grade.

#### Guidelines for collaboration and use of resources

I encourage you to collaborate! Your classmates will be your most valuable resource throughout graduate school. Talk to them, learn from them. But please follow the rules below when it comes to your homework assignments.

Likewise, it is good to look for other perspectives on the material we are studying. Learning math is a continuous process of reorganization in search of an intuitive perspective. Everyone is different and you may find that some texts speak to your intuition better than others.

That said, it is easy to find solutions to every exercise in Dummit and Foote online. Here is a
link. It is easy to collaborate or to use online
solutions in an unproductive or counterproductive way so while you *may* consult other resources,
you must do so according to the rules below:

- Before discussing a problem or consulting an external resource, make an honest effort to solve it yourself.
- Make sure that when you talk to others or read outside material you are actively engaged in producing the solution and learning how it works. One way to do this is to try to improve the solution, either in the mathematical argument or its exposition.
- When you write up your solutions
**do not use any notes or other resources**other than your newly improved brain. This is the true test of whether you have improved your understanding. - You must list any discussions or resources that have contributed to your solutions in any way in the references section of your solutions.
**Failure to do this is plagiarism!** - Anything you submit as
*your*solution to an exercise must be reflection*your*understanding of that exercise and yours alone.

#### Week 1 (§1.1—1.5): Due Monday, 8/31

§1.1: #5, 7, 20, 22, 24, 30 (see Example 6 on p. 18), 32

§1.2: supplement, #3 (try to do this without writing down the multiplication table), 7, 12

§1.3: #5, 7, 11, 15 (the hint is to use Exercise 10 of §3 and Exercise 24 of §1), 16

§1.4: #10

#### Week 2 (§1.6, 1.7, 2.1): Due Wednesday, 9/9

§1.6: #4, 6 (Hint: Look forward to §1.6, Ex. 21), 17, 23, supplement

§1.7: #8, 10a, 17, 18, 19, 21, 23

§2.1: #6 (Hint: consider the "infinite dihedral group"), 8, 10

#### Week 3 (§2.1—2.4): Due Tuesday, 9/15

§1.3: supplement

§1.7: #13

§2.2: #7, 10, 14 (you can assume that F = ℝ if you are unfamiliar with fields)

§2.3: #9, 12b, 18, 24, 25 (note: this map may not be a homomorphism!), 26

§2.4: #7, 12 (Writing out a multiplication table gets no credit for 7 and 12. Hint: use a presentation of the dihedral group.), 14cd, 15

#### Week 4 (§2.5, 3.1): Due Monday, 9/21

§2.5: #14 (just make the diagram; no justification required; you can hand in the diagram on paper if you want)

§3.1: #9, 12, 14, 19, 22, 25, 34, 35, 36, 42, supplement

#### Week 5 (Appendix I, §3.2—3.4): Due Monday, 9/28

§2.4: #17

§3.2: #4, 9, 11, 14, 18, 22

§3.3: #4 (Hint: use universal properties!), 6, 7, 10

§3.4: #4

#### Week 6 (§3.4, 3.5, 4.1, 4.2): Due Monday, 10/5

§3.4: #5

§3.5: #10, 12

§4.1: #1, 9

§4.2: #7 (Hint: Every nontrivial subgroup of Q_{8} contains 〈-1〉), 8, 13 (Hint: compose the left regular permutation representation with the sign homomorphism and compute the image of an element of order 2)

§4.3: #17 (Hint: recognize D as the kernel of an action), 29

#### Week 7 (§4.3, 4.6): Due Monday, 10/12

§4.3: #13, 19, 25, 26 (Hint: use that if H is a subgroup of G and G is a union of conjugates of H then G = H), 30

§4.4: #14

§4.6: #4

#### Week 8 (§4.4, 4.5, 5.1, 5.2): Due Monday, 10/19

§4.4: #1, 20(a)

§4.5: #16, 19, 30, 32, 34

§5.1: #11

§5.2: #13 (observe that you are proving a universal property!)

#### Week 9 (§5.1—5.5): Due Monday, 10/26

§4.4: #18

§5.2: #4bc, 14

§5.4: #5 (remember the universal property of the quotient by the commutator), 7 (this is good prelim practice), 11

§5.5: #11, 22, 23

#### Week 10 (§5.2, 6.3, review): Due Wednesday, 11/4

Do the exam problems as homework. If you are confident you got a problem correct, feel free to skip it and do one of these problems instead.#### Week 11 (§7.1—7.3): Due Monday, 11/9

Do up to 5 of these: §7.1, #3, 5, 6, 7, 15, 21; §7.2, #3b; §7.3, #2, 11

Do these: §7.1, #10, 13bc; §7.2, #2; §7.2, #3c, 5a, 7, 13; § 7.3, #12, 15

Do 5 less the number of problems you did from the first group: §7.1, #26, 27; §7.2, #5b; §7.3, #14, 26c

#### Week 12 (§7.4—7.6): Due Wednesday, 11/18

Do up to 5 of these: §7.3, #5 (remember all ring homomorphisms are unital), 7, 10, 22, 24; §7.4, #9, 15, 18; ~~§7.5, #7~~

Do these: §7.3, #29; §7.4, #2, 13 (for part (a), note that φ^{-1}(P) = R is impossible when φ is a unital homomorphism), 19, ~~39~~, **36**; ~~§7.5, #6, 9~~;

§7.5, #5

Do 5 less the number of problems you did from the first group: §7.3, #33; ~~§7.4, #33, 45, 46 (these three exercises requires a little topology)~~, **§7.4, #40**; ~~§7.5, #11~~

#### Week 13 (§7.6, §8.1—8.2, §9.2): Due Wednesday, 12/2

Do up to 3 of these: §7.6, #1, 7; §8.1, #7, 11b; §8.2, #2, 6a; supplement

Do these: §7.6, #6, supplement; §8.1, #4 (for the first half of part (a), you do not need R to be a Euclidean domain), 6 (problem 4 may be helpful), 8

Do 3 less the number of problems you did from the first group: §7.6, #8, 10, 11, supplement

#### Week 14 (§9.4—9.5): Due Monday, 12/14

Do 8 problems:

§9.4, #6, 11

§9.5, #1, 4, 5

supplement

## Exams

There will be two exams, one midterm and one final. Each will be designed to mimic the conditions of the preliminary exam. Each will have three questions, either drawn from previous preliminary exams or in the same style as problems from previous exams.

- Midterm exam: Thursday, October 29, 6—8pm in ECCR 110
- Final exam:

## Policies

Notify me if you need to miss an assignment or exam. I'm happy to make accommodations as long as your request is reasonable. The more advance warning you give me about your needs, the easier it will be to accommodate them.

## Other materials

Groups of order 90 are not simple