MATH308

MTH308: Groups and fields

[Cr: 4, Lc: 3, Tt: 1, Lb: 0]

Sorry to spam t

Announcements:

Final Exam is on 27/4/18, Friday, 9:00-12:00 in LH3.

Course Outline:

• • Recapitulation: Cayley's Theorem, Class equations, Group of prime power order.

  • Sylow's Theorems, Groups of order < 9.

  • Free groups, generators and relations, commutators, and derived and central series of subgroups.

  • Free Abelian groups, further examples of groups of small order.

  • Field extensions, algebraic extensions, perfect fields, separable and normal extensions.

  • Finite fields, algebraically closed fields.

  • Automorphisms of extensions, cyclotomic extensions.

  • Galois extensions and Galois group, fundamental theorem of Galois theory.

  • Solutions of polynomial equations by radicals, constructible numbers.

Additional Topic

• • Infinite Galois extensions.

Pre-requisites: I shall assume that the people have done a basic course of group theory. For instance they should know homomorphisms, normal subgroups, quotient groups, and basic theorems on these concepts like the Lagrange's theorem. I shall also assume basic knowledge of rings, ideals and quotient rings, polynomial rings over commutative rings, integral domains and fields etc. I will be able to recall some of these concepts very briefly as and when needed. Students not comfortable with any of these things should revise these on their own or ask me during office hours. The prerequisite is the content of Chapter 1,2,3,7,10 and 11 of the text book (see below).

Text books: Dummit and Foote's Abstract Algebra will be used as the textbook. Hungerford's Algebra will be used as a reference.

Homework: At most two homework sets will be assigned every week. It will be posted in this web page. There will be starred homework problems which are supposed to be more important than others for understanding the material covered in class. No HW will be collected or graded.

Exams: There will be two midterms and one final exam. The division of marks for the exams and the HW is to be found below. Exam dates will not be changed for any particular student unless he/she has some sort of provable emergency due to which the student is unable to write the exam on the due date.

Quizzes: I will hold a few surprise quizzes during the semester. Note that quizzes will be graded. If you are absent on that day you miss it and get a zero for the same.

Grades:

Quizzes: 20%

Midterm 1: 20%

Midterm 2: 20%

Final: 40%

Links to notes on classes and HW is to be found below.