Groups, Rings, and Fields

Math 620
Fall 2024
Meeting Mo., We., Fr., 9:00 - 10:00.
GMCS 307
San Diego State University
First Class: Mo. Aug. 26
Last class: Mo., Dec. 11.
Final Exam: Mo., Dec. 16, 8:00-10:00.

Professor: Mike O'Sullivan
Web page: http://mosullivan.sdsu.edu
Email: mosullivan@sdsu.edu
Office: GMCS #582
Office Hours: Mo., We., Fr., 10:00-11:30.
You may make an appointment for another time, or just stop by my office. If I am in and available, we can talk.

Detailed Information


Syllabus

My lecture notes, Ch 1, 2
My lecture notes, Ch 3
My lecture notes, Ch 4

SCHEDULE
ASSIGNMENTS

Course Description

We will study the core objects in abstract algebra: groups, rings and fields. The term ``abstract'' highlights the approach to mathematics that emerged in the 20th century in which there was less reference to ``real objects'' and a greater emphasis on proving theorems from a small set of simple axioms. For example, the symmetries of a 3-dimensional object tell you something about the nature of the object. But, symmetry itself is interesting. Symmetries of an object can be composed (apply one then another) and produce another symmetry. Each symmetry has an inverse, which undoes the given symmetry. Abstractly, one can consider the properties of a set (e.g. symmetries) with an operation (e.g. composition) and an inverse to the operation. We call such an object a group. Discovering the structure and properties of abstracts groups was a massive project in the last century.

Ring theory evolves from recognizing that the basic properties of the integers (from grade school!)---two operations (+, *), with commutativity, associativity, distributivity---extend to polynomials and can be extended further to abstractly defined objects. Similarly, the properties of the rational numbers (now every nonzero element has a multiplicative inverse) can be considered in an abstract axiomatic way. This is the area called field theory.

I love this material because it gives a deeper and richer understanding of school mathematics, but also unveils a world of very complex and intricate beauty.

Resources

O'Sullivan, Groups, Rings, and Fields Course notes in progress!
Required.
Hungerford, Abstract Algebra: An Introduction 2nd ed.
(Recommended) This is the textbook we use for our undergraduate algebra courses. It is well written and should be a nice reference to flesh out details and give examples.
David Dummit, Richard Foote, Abstract Algebra. A massive standard reference for graduate level algebra, full of examples and detailed proofs.
Gallian, Contemporary Abstract Algebra.
(Alternative possibility) A widely used and well written text that can also be a reference.
Judson and Beezer, Abstract Algebra: Theory and Applications
(Alternative possibility) This is another good undergraduate algebra text. Available free online. See the button for "Sage and AATA." Download "Sage Worksheet Collection." Open the zip file from the sage notebook.
Ash, Robert, Abstract Algebra: The basic graduate year .
Available online.
SDSU Sage Tutorial Updated 2019. Sage Tutorials Magma online calculator.

Prerequisites

Some experience with abstract algebra: group theory, ring theory, and number theory. If you have just studied one of these topics, or did not obtain or grade of B or better, be prepared to work a bit harder (allot time!) in the course.

Primary Topics

Some of the following topics will be familiar from your undergraduate course. We will cover them in greater depth, and with more attention to details. I expect this material to take 9 weeks.

Groups: Normal subgroups and the isomorphism theorems. Finite abelian groups, semi-direct products, simple groups, group presentations.
Commutative Rings: Ideals, homomorphisms, and quotient rings of polynomial rings over a field. Modules.
Fields: Field extensions. Adjoining a root of a polynomial. Splitting field of a polynomial. Finite fields.

Format

Class time will mix lecture with problem solving. We may also spend some time using SageMath or Magma in the computer lab. I will assign specific pages from my notes to be read before class. You may not understand some of the material, but, read the assigned pages, formulate questions that you have, and be prepared to discuss this in class. I have some short (5-15 min.) recorded lectures on some topics, which will free class time for discussion of problems. Be prepared to present your work in class and also to work on problems in class.

Learning Outcomes

It is standard these days to have learning outcomes for every course; rather than simply listing the topics covered. My approach to this is as follows. In every math course that I teach, I want students to advance in the skills listed below (adapted from the Degree Learning Outcomes for the SDSU math major as presented on the department website). In this course we do this work in the context of groups, rings and fields.

[Foundational knowledge.] State major definitions, axioms, and theorems and use examples to illustrate.
[Use logical reasoning.] Read a proof and explain the logic and derivations. Write a mathematical proof using an appropriate method.
[Use algebraic tools and methods.] Derive answers, apply algorithms, and compute, both by hand and using mathematics software.
[Explore mathematical ideas independently.] Have confidence to read challenging material that is beyond that explored in a textbook or class.
[Communicate mathematical ideas effectively.] Make progress toward the mathematicians goal: writing that gets to the essence of the matter and is brief, clear, and polished.
[Nurture the learning of others.] Work with others in a way that is collegial, inclusive and empowering. Contribute, but seek understanding of other perspectives.