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2016　Algebra III

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Kato Fumiharu 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Tue5-6(H136)  Fri5-6(H136)  
Group
-
Course number
MTH.A331
Credits
2
Academic year
2016
Offered quarter
3Q
Syllabus updated
2016/4/27
Lecture notes updated
-
Language used
Japanese
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Course description and aims

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.

Student learning outcomes

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 algebraic closure of fields, 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 that will be developed

✔ 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/Required learning

  Course schedule Required learning
Class 1 introduction, notion of the field Details will be provided during each class session
Class 2 extension of field Details will be provided during each class session
Class 3 simple extension of field, algebraic extension of field Details will be provided during each class session
Class 4 (minimal) decomposition field Details will be provided during each class session
Class 5 algebraic closure of field Details will be provided during each class session
Class 6 separable and inseparable entensions Details will be provided during each class session
Class 7 isomorphisms of fields with their extensions Details will be provided during each class session
Class 8 normal extension, Galois extension, Galois group Details will be provided during each class session
Class 9 fundamental theorem of Galois Theory Details will be provided during each class session
Class 10 calculations of various examples of Galois groups Details will be provided during each class session
Class 11 cyclotomic field Details will be provided during each class session
Class 12 trace and norm, finite field Details will be provided during each class session
Class 13 cyclic Kummer extension Details will be provided during each class session
Class 14 applications of the Galois Theory: solvability of algebraic equations using the roots of unity Details will be provided during each class session
Class 15 applications of the Galois Theory: geometrical construction Details will be provided during each class session

Textbook(s)

Katsura, Toshiyuki, Algebra III, UP of Tokyo

Reference books, course materials, etc.

Artin, Emile, Introduction to the Galois Theory, (Japanese translation), Chikuma Pub, 2010
Nakajima, Sho-ichi, Algebraic Equations and the Galois Theory , Kyoritsu Pub ,2006.

Assessment criteria and methods

Final exam and discussion sessions. Details will be announced during the course.

Related courses

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

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed Algebra I and II.

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