Course title
V0140300
Algebra 3

 shimizu kenichi Click to show questionnaire result at 2017
Course description
The Galois theory describes the structure of a field extension in terms of its group of automorphisms. The theory is important in coding theory, cryptographic theory, random number generation, etc. This course introduced basics on the Galois theory and its application to the solvability of algebraic equations.
Purpose of class
It is an important method to study the structure of a mathematical object via its group of symmetry -- the automorphism group --. The aim of this course is to understand such a method and learn their applications in mathematics.
Goals and objectives
  1. Understand the basics on field extensions
  2. Understand the fundamental theorem of Galois theory
  3. Understand the relation between the Galois theory and the solvability of algebraic equations
Language
Japanese
Class schedule

 Class schedule HW assignments (Including preparation and review of the class.) Amount of Time Required
1. Symmetry of equations Review linear algebra, Algebra I and II. 180minutes
2. Field extensions Review commutative algebras and related topics 180minutes
3. Structure of simple extensions Review the fundamental theorem on homomorphisms 180minutes
4. Degree of extensions (1) Definition and properties Review the notion of basis (from linear algebra) 180minutes
5. Degree of extensions (2) Examples Review the last lecture 180minutes
6. Compass-and-straightedge construction Review the last lecture 180minutes
7. Mid-term exam Review the content of this course 270minutes
8. Symmetry of field extensions Review the fundamentals on group theory (Algebra I) 180minutes
9. Primitive element theorem Review the last lecture 180minutes
10. The fundamental theorem of Galois theory (1) The statement of the theorem Review the last lecture 180minutes
11. The fundamental theorem of Galois theory (2) The proof of the theorem Review the last lecture 180minutes
12. Solvability of algebraic equations (1) Properties of radical extensions Review the fundamental theorem of Galois theory 180minutes
13. Solvability of algebraic equations (2) Abel-Ruffini theorem Review the last lecture 180minutes
14. Final exam Review the content of this course 270minutes
Total. - - 2700minutes
Relationship between 'Goals and Objectives' and 'Course Outcomes'

 Mid-term exam Final exam Report Total.
1. 10% 10% 10% 30%
2. 10% 10% 10% 30%
3. 10% 10% 20% 40%
Total. 30% 30% 40% -
Evaluation method and criteria
Evaluate by the scores of the mid-term exam, the final exam and reports.
Textbooks and reference materials
Reference materials will be announced in the first lecture.
Prerequisites
Linear Algebra I, II, Linear Space, Algebra I, and Algebara II are assumed
Office hours and How to contact professors for questions
  • 12:30-13:10 of Monday, or anytime I'm in the lab.
Relation to the environment
Non-environment-related course
Regionally-oriented
Non-regionally-oriented course
Development of social and professional independence
  • Non-social and professional independence development course
Active-learning course
N/A
Last modified : Wed Oct 17 07:32:43 JST 2018