Algebra III

Academic unit or major

Undergraduate major in Mathematics

Instructor(s)

Yuichiro Taguchi

Class Format

Lecture

Media-enhanced courses

-

Day of week/Period
(Classrooms)

5-6 Tue (H137) / 5-6 Fri (H137)

Class

-

Course Code

MTH.A331

Number of credits

200

Course offered

2021

Offered quarter

3Q

Syllabus updated

Jul 10, 2025

Language

Japanese

Syllabus

Course overview and goals

The main theme of this course is Galois Theory, based on the theory of finite field extensions, and its various applications. Galois Theory is one of the most important theories in modern algebra, giving foundational approach to modern mathematics, and, at the same time, one can say, one of the final subjects in the undergraduate algebra course.
In this course, we learn the basics of Galois Theory and its applications, including the solvability of algebraic equations and geometrical construction.

Course description and aims

Students are required to learn the basics of the theory of finite field extensions, including the construction of finite extension field via the residue fields, by maximal ideals, of the polynomial ring. After learning the basics including the existence of an algebraic closure of a field, we proceed to Galois Theory, such as the Galois correspondence between subgroups of the Galois group and fixed fields, of which the students are required to have good understanding. Also required is to understand its applications, such as finite fields, the solvability of algebraic equations, and geometrical construction.

Keywords

Galois extension, fundamental theorem of Galois theory, finite field, solvability of algebraic equations

Competencies

• Specialist skills
• Intercultural skills
• Communication skills
• Critical thinking skills
• Practical and/or problem-solving skills

Class flow

Standard lecture course accompanied by discussion sessions

Course schedule/Objectives

Study advice (preparation and review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

None required

Reference books, course materials, etc.

E. Artin, "Galois Theory", Dover
N. Bourbaki, "Algebra II", Springer-Verlag
T. Szamuely, "Galois groups and fundamental groups", Cambridge University Press

Evaluation methods and criteria

By exams and reports. Details will be announced in the course.

Related courses

• MTH.A301 ： Algebra I
• MTH.A302 ： Algebra II

Prerequisites

Students are expected to have passed Algebra I and II.