2025 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
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) - Class
- -
- Course Code
- MTH.A331
- Number of credits
- 200
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
The main theme of this course is the theory of separable algebras of finite degree over a field, Galois theory, and their applications. Galois theory and its variants play a fundamental role in modern mathematics. In this course, students are expected to understand the fundamental theorem of Galois theory as well as various applications of it such as the solvability criterion for algebraic equations.
Course description and aims
To understand the basic theory of finite separable algebras over a field, in particular of finite Galois extensions of a field.
To understand the fundamental theorem of Galois theory, which describes the correspondence between finite separable algebras (and their morphisms) and finite sets furnished with a group action (and their morphisms).
To understand applications of Galois theory such as Kummer theory, the solvability criterion for algebraic equations, the theory of finite fields and of cyclotomic extensions.
Keywords
finite separable algebra, Galois extension,Galois group, fundamental theorem of Galois theory, solvability of algebraic equation
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 | Finite algebras over a field | Details will be provided during each class session |
| Class 2 | Dedekind's theorem | Details will be provided during each class session |
| Class 3 | Separable algebras | Details will be provided during each class session |
| Class 4 | Artin's theorem | Details will be provided during each class session |
| Class 5 | Fundamental theorem of Galois theory | Details will be provided during each class session |
| Class 6 | Traces and norms | Details will be provided during each class session |
| Class 7 | Galois cohomology | Details will be provided during each class session |
| Class 8 | Kummer theory, Artin-Schreier theory | Details will be provided during each class session |
| Class 9 | Splitting fields, algebraic closure | Details will be provided during each class session |
| Class 10 | Separable algebraic extensions, normal extensions | Details will be provided during each class session |
| Class 11 | Galois group of a polynomial, solvability of an algebraic equation | Details will be provided during each class session |
| Class 12 | Finite fields | Details will be provided during each class session |
| Class 13 | Cyclotomic extensions | Details will be provided during each class session |
| Class 14 | Further topics | Details will be provided during each class session |
Study advice (preparation and review)
Students are expected to study the relevant parts of the handouts before each of the classes.
Textbook(s)
none in particular
Reference books, course materials, etc.
T. Szamuely, Galois Groups and Fundamental Groups, Cambridge U.P.
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
It is desirable that students have taken "Algebra I" and "Algebra II".