• Mapping the Mind

    Junior computer science and mathematics double major Brooke Guo analyzes neural connections to understand the causes of complex brain conditions like schizophrenia.  When Brooke Guo arrived at the University of Maryland as a freshman in 2022, she knew she wanted to help people and work in a health-related field someday.… Read More

  • Four Science Terps Awarded 2025 Goldwater Scholarships

    Four undergraduates in the University of Maryland’s College of Computer, Mathematical, and Natural Sciences (CMNS) have been awarded 2025 scholarships by the Barry Goldwater Scholarship and Excellence in Education Foundation, which encourages students to pursue advanced study and research careers in the sciences, engineering and mathematics.  Over the last 16 years, UMD’s nominations… Read More

  • 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

  • 2024 Putnam Results

    We are very excited to report that our MAryland Putnam team ranked 7th among 477 institutions that participated in the 2024 Putnam math competition. Our team members this year were Daniel Yuan, Isaac Mammel, and Clarence Lam. Daniel Yuan ranked 26th among 3,988 participants. Clarence Lam and Isaac Mammel were recognized for… Read More

  • From Math Olympiads to Diplomacy: Meet Visiting Math Professor Qendrim Gashi

    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 course discusses the basic elementary facts about the integers, groups, rings, integral domains, and fields. It is recommended strongly for all students planning to go to graduate school in mathematics. 

Prerequisites

1 course with a minimum grade of C- from (MATH240, MATH461, MATH340); and 1 course with a minimum grade of C- from (MATH341, MATH241); and minimum grade of C- in MATH310.


Level of Rigor

Advanced


Sample Textbooks

Contemporary Abstract Algebra, by J. Gallian

Abstract Algebra, by I.N. Herstein


Applications

Chemistry, Engineering, Physics, and Astronomy


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

MATH406, MATH456, MATH404, MATH405


Additional Notes

Duplicate credit with MATH402

Students interested in grad school in MATH should strongly consider this course

Topics

Groups

Definition and examples

Subgroups

Normal subgroups and quotient groups

Homomorphisms

Isomorphism theorems

Cyclic groups

Permutation groups, Cayley's Theorem

Rings

Definition and examples

Polynomial rings

Ideals and quotient rings

Integral domains

Euclidean domains, unique factorization

Prime ideals, maximal ideals

Fields

Quotient fields

Extensions

Algebraic extensions

Impossibility of trisection of angle

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