MAS 4302 (19258) – Abstract Algebra 2 – Spring 2025

  • Instructor: Dr. Richard Crew
  • Place and time: LIT 221, MWF 4
  • Office Hours: LIT 404, time TBD, or by appointment (see below)

Course Description

This is a course on the theory of rings, fields and Galois theory. The text is Ian Stewart’s Galois Theory, 4th edition but the 3rd edition may be used (with appropriate care). We may make some use of Gallian’s Contemporary Abstract Algebra which was the text for MAS 4301, but this will be limited mainly to some basic ring theory. After a historical introduction and review of some ring theory, we will take up the theory of field extensions, ruler-and-compass constructions and the basic results of Galois theory, culminating in Galois’ theorem on the solvability of equations by radicals. If time permits we will study the Galois theory of finite fields and Gauss’s construction of the regular 17-gon. This may seem like a rather disparate collection of topics, but they are all based on the same underlying theme: “what does it mean to have a particular type of algebraical formula for the roots of an equation”? The underlying structure here is a group-theoretic one; in fact this very question was the reason why groups were studied in the first place.

It will be assumed that students have a basic competence in reading and writing proofs, which has been developed in the previous courses. Lectures will mostly be devoted to discussion of the proofs in the text.

Requirements

The sole prerequisite is MAS 4301, Abstract Algebra 1. This is of necessity, since this course is about an application of group theory! Less formally, Less formally, it will be assumed that students have a basic competence in reading and writing proofs, which has been developed in the previous courses. Apart from this there are no special requirements, other than a sense of adventure.

Lectures and office hours

Lectures will mostly be devoted to discussion of the proofs in the text, but a significant amount of time will be devoted to discussion. For this reason it is essential that you read the relevant sections of the text before the lecture. If you need to see me and cannot attend my scheduled office hours, please email me to arrange for a meeting. If you have any questions about the assigned work, do not hesitate to ask! This is what the classroom sessions and office hours are for. You can also email questions to me, but after 5pm please don’t expect an immediate answer. If you send me a question by email, please note that I may not have a copy of the text with me. In particular if you have questions about a homework problem, be sure to state the problem in the email.

Software

There will be no officially assigned computer work in this course but some computer work is often enlightening and fun. Perhaps the best software suite for this course is SageMath, which is a free and open source software system for all kinds of algebraic computation. It runs on Windows and most varieties of Unix (Linux, Mac) and has a scripting language based on Python. It can be downloaded from the SageMath site, which has complete installation instructions.

Quizzes, Exams and Homework

The course grade will be based on 4 exams (each worth 1/6 of the final grade), a number of quizzes and assigned homework (in total, the remaining 1/3 of the grade). Dates for all homework, quizzes and exams will appear on the CANVAS pages for this course. Automatic makeups for quizzes and exams will be given for documented emergencies, curricular requirements of other University units, military service or mandatory Court appearances. In other situations, late submission of homework and makeups for quizzes and exams will be allowed if arrangements are made at least two days in advanced and the reasons are plausible.

Dates of exams and quizzes may be changed due to extreme weather or other natural disasters. These will
be announced in class, but you should consult the CANVAS pages for this course if you have any questions.

Grading Scale

Grades: A=90-100, A-=87-89, B+=83-86, B=78-82, B-=75-77, C+=70-74, C=65-69, C-=60-64, D=50-59, E=0-49. The UF regulations on grades are here. The UF policy on minus grades is here.

Course Policies

Attendance in this course is not mandatory, but for the reasons stated above you will want to get notes from your fellow students if you have to miss a class. In general the more class discussions you attend, the better will be your performance on quizzes and exams. This policy is consistent with the UF policy on attendance.

UF students are bound by The Honor Pledge which states, “We, the members of the University of Florida community, pledge to hold ourselves and our peers to the highest standards of honor and integrity by abiding by the Honor Code. On all work submitted for credit by students at the University of Florida, the following pledge is either required or implied: “On my honor, I have neither given nor received unauthorized aid in doing this assignment.” The Honor Code specifies a number of behaviors that are in violation of this code and the possible sanctions. Furthermore, you are obliged to report any condition that facilitates academic misconduct to appropriate personnel. If you have any questions or concerns, please consult with the instructor of this class.

That said, discussion of assigned work with your fellow students is always allowed, or in other words will count as “authorized aid” for the purposes of this course. In fact forming discussion groups and discussing problems is one of the best ways to learn this (or any) mathematical subject. It is how professional mathematicians operate. Nonetheless, the work that you turn in must be your own.

Students are expected to provide professional and respectful feedback on the quality of instruction in this course by completing course evaluations online via GatorEvals. Guidance on how to give feedback in a professional and respectful manner is available here. Students will be notified when the evaluation period opens, and can complete evaluations through the email they receive from GatorEvals, in their Canvas course menu under GatorEvals, or else here. Summaries of course evaluation results are available to students here.

Accomodations

Students requesting accommodations for disabilities should first register with the Disability Resource Center, (352-392-8565) by providing appropriate documentation. Once registered, students will receive an accommodation letter which must be presented to the instructor when requesting accommodation. Students with disabilities should follow this procedure as early as possible in the semester.

Health and Wellness

U Matter, We Care: If you or a friend is in distress, please contact umatter@ufl.edu or 352-392-1575 so that a team member can reach out to the student. Other resources:

Metadata

This page is the official syllabus for this course, and will be modified as needed. Last modified on Jan 1, 2025.