The Algebra Qual

Welcome to the unofficial home for all things related to the Qualifying Exam in Algebra (aka "The Algebra Qual") at California Polytechnic State University, San Luis Obispo. The Algebra Qual consists of five problems covering three main subfields of abstract algebra, namely:

• linear algebra
• group theory
• ring theory

See the exam syllabus for a detailed breakdown of the topics covered within each of the above subfields.

Each problem is graded for correctness and clarity of presentation, and is worth at most five points. A passing score is 15 points (out of 25).

Problem bank

Beginning with the September 2025 exam, all problems will be drawn from a public problem bank. This bank contains two types of problems: template problems and pool problems.

Template problems are generally computational with easily adjustable specifics. These types of problems are especially prevalent in linear algebra. Examples include problems such as:

• analyzing a geometrically-given linear transformation for specific properties, such as eigenvalues or diagonalizability; e.g., see March 2025 (Problem 5) or Spring 2019 (Problem 1)
• counting the number of morphisms between two specific groups; e.g., see Spring 2017 (Problem 4) or Fall 2015 (Problem 1)

Pool problems make up the rest of the problem bank, and include all problems that are not easily adjustable. These problems, when chosen, will usually be asked as is.^[1]

For additional information about the problem bank, including links to the actual problems, click here.

Historical information

I don't know exactly when this exam officially came into existence, but the earliest exam to which I currently have access is from 1988. The following historical notes refer to all exams since 1988.

• There has always been at least one problem in each topic.
• Since June of 2005 there have always been at least two problems in group theory.
• Exactly six exams contain only a single problem in group theory, namely the exams on December 1988, March 1992, March 2000, November 2000, March 2001 and March 2005.
• Only two exams have contained three problems on the same topic. The May 1991 exam contained three problems in group theory, while the March 2001 exam contained three problems in ring theory.

If you would like to see these old exams, the official repository is available here. Recent exams have also been recreated here.

Related pages

Exam syllabus
Problem bank
Notation Key
Past exams
Changelog