Course Content

Lecture Notes

Notes by Mars Semenova

Homework assignments

• Assignment 1, due 28 January 2025.
• Assignment 2, due 6 February 2025.

Exams

The midterm exam will be held in class on February 13. The final exam will be April 17 at 3:30pm.

• Sample Midterm 1 and solutions.
• Sample Midterm 2 and solutions.
• Sample Final 1 and solutions.
• Sample Final 2 and solutions.

Approximate agenda

1. Rings: definitions and examples.
2. Homomorphisms, ideals, and quotients.
3. Prime and maximal ideals.
4. Fractions.
5. Euclidean domains. Examples and nonexamples.
6. Principle ideal domains and unique factorization domains.
7. Polynomials over UFD are a UFD.
8. Irreducibility criteria.
9. Noetherian rings and the Hilbert basis theorem.
10. Gröbner bases.

References

• David S. Dummit, Richard M. Foote, Abstract Algebra, 3rd Edition, 2003.
• Tom Judson, Abstract Algebra: Theory and Applications, 2022.
• David A. Cox , John Little , Donal O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2015.
• John B Fraleigh, First Course in Abstract Algebra, 2020.
• Serge Lang. Algebra, 2002.

Fun exercises

Mike Pierce, at UC Riverside, maintains lists of fun exercises in various topics, targeted at their Qualifying Exam. (In the UC system, qualifying exams are typically taken upon arrival the first week of starting graduate school. Students are required to pass by the end of their first year.) These same questions make good practice questions for anyone studying the subject. I particularly like his list of Ring Theory questions; there are also some good questions on his Commutative Algebra list.