2023 Faculty Courses School of Science Undergraduate major in Mathematics
Algebra III
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Yuri Yatagawa
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Tue (M-B45(H105)) / 5-6 Fri (M-B45(H105))
- Class
- -
- Course Code
- MTH.A331
- Number of credits
- 200
- Course offered
- 2023
- Offered quarter
- 3Q
- Syllabus updated
- Jul 8, 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
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Fields and their extensions | Details will be provided during each class session |
| Class 2 | Simple extensions, algebraic extensions | Details will be provided during each class session |
| Class 3 | Algebraic closure of a field | Details will be provided during each class session |
| Class 4 | Separable and inseparable extensions | Details will be provided during each class session |
| Class 5 | Isomorphisms of fields, extensions of an isomorphism | Details will be provided during each class session |
| Class 6 | Minimal decomposition fields, normal extensions | Details will be provided during each class session |
| Class 7 | Galois extensions and Galois groups | Details will be provided during each class session |
| Class 8 | Fundamental theorem of Galois Theory | Details will be provided during each class session |
| Class 9 | Calculations of various examples of Galois groups | Details will be provided during each class session |
| Class 10 | Cyclotomic fields | Details will be provided during each class session |
| Class 11 | Trace and norm. Finite fields | Details will be provided during each class session |
| Class 12 | Cyclic Kummer extensions | Details will be provided during each class session |
| Class 13 | Application of Galois Theory: Solvability of algebraic equations | Details will be provided during each class session |
| Class 14 | Application of Galois Theory: Drawing with ruler and compass and examples | Details will be provided during each class session |
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)
T. Yukie "Algebra 2" Nippon Hyoron sya
Reference books, course materials, etc.
T. Katsura. "Algebra III", University of Tokyo Press
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.