Galois Theory

Course Overview

This course introduces undergraduate students to one of the most beautiful achievements of modern algebra: Galois theory, the study of symmetries of polynomial equations. Starting from classical questions about solving equations by radicals, we will develop the language of groups and fields, culminating in the fundamental theorem of Galois theory. Along the way, students will see how abstract algebra connects with concrete problems. This course is aimed at undergraduates with an interest in algebra and number theory. No prior exposure to Galois theory will be assumed.

Course Information

• Instructor: Mihran Papikian (Professor of Mathematics, Pennsylvania State University). With any questions regarding the course you can contact the instructor by email. 

• Location: The auditorium of the Institute of Mathematics of the Academy of Sciences (24/5 Marshal Baghramyan Avenue, Yerevan).

• Duration: 12 weeks, starting September 16. 

• Schedule:  Tuesday and Friday every week, 18:00-21:00.

• Lecture: Each lecture will be 90 minutes long, followed by a 60-minute recitation (practice session). 

• Mode: The instructor will join and deliver the lectures over zoom. The recitations will be taught by the course TA (Hayk Karapetyan) in person.

• Materials: Lecture notes will be provided weekly.

• Assessment: Weekly problem sets, a written midterm exam, and an oral final exam.

Prerequisites

• Basic knowledge of English. The lectures will be in Armenian, but the lecture notes and homeworks will be in English. 

• Familiarity with basic linear algebra.

• Mathematical maturity and readiness to work with proofs.

• Some familiarity with the theory of groups and fields will be helpful but not necessary.

Topics

• Historical Motivation: The problem of solving polynomial equations.

• Field Extensions: Basic definitions, minimal polynomials, and algebraic extensions.

• Automorphisms and Symmetries: Groups of field automorphisms.

• Splitting Fields and Normal Extensions.

• The Fundamental Theorem of Galois Theory.

• Applications: Solvability of equations by radicals; classical impossibility results (e.g., trisecting an angle, doubling the cube, constructing regular polygons); modern perspectives.