Abstract Algebra

A four-course, proof-based sequence in the structures underlying modern mathematics: groups, rings, fields, and Galois theory, ending with why the quintic has no formula in radicals.

Group Theory

Introduction to Group Theory is a proof-based course exploring symmetry and algebraic structure through rigorous problem solving and mathematical exposition. Topics include groups and subgroups, symmetric and dihedral groups, cyclic groups and orders of elements, cosets and Lagrange’s theorem, homomorphisms and isomorphisms, quotient groups, group actions, Burnside’s lemma, and the Sylow theorems.

Subgroups
Symmetric & dihedral groups
Lagrange’s theorem
Homomorphisms
Quotient groups
Group actions
Burnside’s lemma
Sylow theorems

Ring Theory

This course introduces the algebraic structure of rings, exploring how familiar properties of the integers and polynomials extend to more general settings. Topics include ideals, quotient rings, ring homomorphisms, integral domains, polynomial rings, and unique factorization. Students develop a deeper understanding of algebraic structures through rigorous proofs and problem solving.

Ideals
Quotient rings
Ring homomorphisms
Integral domains
Polynomial rings
Unique factorization

Field Theory

This course explores the structure of fields and their extensions, with an emphasis on understanding polynomial equations and their solutions. Topics include field extensions, algebraic and transcendental elements, minimal polynomials, splitting fields, and finite fields. Students investigate how algebraic structures provide a framework for studying roots of polynomials and connections between different number systems.

Field extensions
Algebraic & transcendental elements
Minimal polynomials
Splitting fields
Finite fields

Galois Theory

This course develops the beautiful correspondence between field extensions and groups of symmetries, culminating in the Fundamental Theorem of Galois Theory. Topics include field automorphisms, splitting fields, Galois groups, and solvability of polynomial equations by radicals. Students explore applications to classical geometric constructions and the impossibility of a general formula by radicals for polynomial equations of degree five or higher.

Field automorphisms
Galois groups
Fundamental Theorem
Solvability by radicals
Geometric constructions
The quintic

Every PiMath course

How classes work

Live online

Seminar-style discussion over Zoom. Every session is recorded.

12 students at most

Small enough that every student is heard and known.

Canvas course site

Schedule, recordings, annotated slides, and course notes in one place.

Feedback on every proof

Detailed comments on problem solving and proof writing.

Ready to apply?

Placement is based on readiness and interest rather than age or grade level. If you are not sure which course fits, choose “Not sure which course fits — please advise” on the application and we will help.

Open the application form Questions? Contact us

Financial aid and scholarships

Partial and full need-based scholarships are available for every course. Ask about aid on the application.

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