## Math 5031 – Algebra I (Fall 2021)

### Section information

**Class time and location**: M-W-F from 11:00AM to 11:50AM in Cupples II, L011**Tentative office hours**: Monday-Wednesday-Thursday 2:00PM to 4:00PM (shared)

### Subject

This is the first half of a two-semester course that provides the basis for the Ph.D. qualifying exam in Algebra. We will cover primarily groups, rings and modules. It is expected that you have had prior exposure to Linear and Abstract Algebra at the undergraduate level (as in Math 429 and 430).

### Text

I plan to follow

**Basic Algebra I**by Nathan Jacobson, second edition, Dover, 2009.

On WebSTAC and the campus bookstore it will say that the textbook for the course is **Algebra** by Larry C. Grove. I have decided to change the textbook to Jacobson’s for the sake of continuity since this is the text that will be used in Math 5032 (Algebra II) by Prof. Matt Kerr. The two books may complement each other since the one by Grove is very concise whereas Jacobson’s text is somewhat more discursive (and more interesting to read) so they balance each other in this sense. But I will not rely in any way on Grove’s text. I plan to follow Jacobson somewhat closely, up to the end of chapter 3 plus some additional topics as time permits. Both texts are relatively inexpensive (Dover editions) and it should not be difficult to find copies online if you can’t get a hard copy in time for the beginning of classes.

In addition to Jacobson’s text, you will likely find Matt’s lecture notes for Algebra I useful. I will likely refer to those notes often in class and in homework assignments. They can be found in his webpage here and through the below links on the list of topics.

### Topics we hope to cover.

The links in the following list of topics are to Prof. Matt Kerr’s notes. I may refer to some of them occasionally in addition to the textbook. You may find Matt’s notes more straightforward to read than Jacobson’s text.

- Sets
- Groups
- Introduction
- Permutation groups
- Groups and subgroups
- Cosets and Lagrange’s theorem
- Homomorphisms and isomorphisms
- Group actions
- Conjugacy and the orbit-stabilizer theorem
- Cauchy’s theorem
- Normal subgroups and quotient groups
- Automorphisms
- Generators and relations
- The Sylow theorems
- Some results on finite groups
- Burnside’s counting lemma

- Rings
- Modules
- Remarks on Associative Algebras

### Coursework and grades

Coursework will consist of homework and reading assignments, two mid-term tests and one final exam. Your final grade is determined as follows: homework 50%, midterms 10% each, final exam 30%. The lowest grade you received on homework will be dropped. Assignments will be handled through Crowdmark and grades will be posted on Canvas.

Grades may be curved, but will not be less than the following scale:

A+ | A | A- | B+ | B | B- | C+ | C | C- | D | F |
---|---|---|---|---|---|---|---|---|---|---|

TBA | 90+ | [85,90) | [80,85) | [75,80) | [70,75) | [65,70) | [60,65) | [55,60) | [50,55) | [0,50) |

If you are a graduate student, a letter grade of B is required to pass; if you are an undergraduate taking this class Pass/Fail, you need a C- to pass.

In writing the tests, I will rely strongly on the homework assignments. To feel confident, you should master the topics and exercises covered by the assignments.

### Schedule of tests and final

Test | date |
---|---|

MT 1 | October 1 |

MT 2 | November 5 |

Final | December 21, 10:30AM - 12:30PM (same place as classes) |

Exams will likely be taken in class. This is something I still need to decide. The above information about the final exam is what is given in WebSTAC; I’m hoping the room will be available to us beyond the 12:30PM limit. I’ll let you know more in class.

### Recording

I plan to record the in-person lectures using Zoom and make them available in Canvas.

### Academic integrity

I will follow the University’s academic integrity policy, which you can read here. Work submitted under your name is expected to be your own. You are permitted, and encouraged, to collaborate on assignments. You may research broadly over the internet and in books when working on assignments, but please indicate your sources. (You won’t lose points for doing this kind of research!)

If you have any concerns or questions about this policy or academic integrity in class, please contact me.

Please include **Math 5031** in the subject line of any email message that pertains to this course.

### Homework assignments

Solutions will be posted in Canvas (under Pages)

- Homework 01: due 09/10/21
- Homework 02: due 09/17/21
- Homework 03: due 09/24/21
- Homework 04: due 10/01/21 (This is also a mid-term exam. More information in class.)
- Homework 05: due 10/08/21
- Homework 06: due 10/22/21
- Homework 07: due 10/29/21
- Homework 08: due 11/05/21 (This is also a mid-term exam. More information in class.)
- Homework 09: due 11/12/21
- Homework 10: due 11/19/21
- Homework 11: due 12/03/21
- Homework 12.