• Mapping the Mind

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  • Announcing the Winners of the Frontiers of Science Awards

    Congratulations to our colleagues who won the 2025 Frontiers of Science Award: - Dan Cristofaro-Gardiner, for his join paper with Humbler and Seyfaddini: “Proof of the simplicity conjecture”, Annals of Mathematics 2024. - Dima Dolgopyat & Adam Kanigowski, for their joint paper with Federico Rodriguez Hertz: “Exponential mixing implies Bernoulli”, Annals of Mathematics… Read More

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    Maryland Global, published a great interview with our visiting professor (and diplomat), Qendrim Gashi. The interview is available at https://marylandglobal.umd.edu/about/news/math-olympiads-diplomacy-meet-visiting-math-professor-qendrim-gashi Read More
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Description

This is the less theoretical of the two upper level undergraduate abstract algebra courses. (Student's planning graduate work in mathematics should take the other course in abstract algebra, MATH 403.) The course covers the basics of groups, rings, integral domains and fields. There will be a detailed study of several groups, and a study of the properties of integers and polynomials. Emphasis is on the origin of the mathematical ideas studied and the logical structure of the subject. (This course is not open to mathematics graduate students.) The last item in the list of topics could be replaced by other topics in algebra, for example, coding theory or crystallographic groups.

Prerequisite:

1 course with a minimum grade of C- from (MATH240, MATH341, MATH461).

 

Level of Rigor

Standard

 

Sample Textbooks

A First Course in Abstract Algebra, by Fraleigh

Contemporary Abstract Algebra, by J. Gallian

 

Applications

Chemistry (Crystallography)

 

If you like this course, you might also consider the following courses

MATH406, MATH456 

 

Additional Notes

Duplicate credit with MATH403

Topics

Preliminaries

Induction

Equivalence relations

Functions

Groups

Groups and symmetries

Finite groups

Cyclic groups

Subgroups

Permutation groups

Cosets and Lagrange's Theorem

External direct product

Normal subgroups and factor groups

Homomorphisms

The structure of finite abelian groups

Rings

Introduction to rings

Integral domains

Ideals and factor rings

Ring homomorphisms

Polynomial Rings

Division Algorithm

Unique Factorization of Polynomials

Divisibility in Integral Domains

Fields

Extension Fields

Algebraic Extensions

Finite Fields

Geometric Constructions

Constructible numbers

Trisecting an angle

Duplicating a cube

Squaring a circle

 

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