Description

Algebraic and transcendental elements, Galois theory, constructions with straight-edge and compass, solutions of equations of low degrees, insolubility of the Quintic, Sylow theorems, fundamental theorem of finite Abelian groups.

Prerequisites

Minimum grade of C- in MATH403. (Math402 does not meet the prerequisite)

Level of Rigor

Advanced

Sample Textbooks

Fields & Galois Theory by Howie

Galois Theory, by I. Stewart

Applications

If you like this course, you might also consider the following courses

Additional Notes

Students interested in grad school in MATH should strongly consider this course 

Topics

Elementary facts about fields

Characteristic

Integral domains

Maximal ideals

Field extensions

Transcendental and algebraic extensions

Liouville numbers

Minimal polynomials

Review of vector spaces

Linear independence

Basis

Dimension

Degree of a field extensions

Straight-Edge and compass constructions

Impossibility constructions

Cyclotomic polynomials

Constructibility of regular n-gons

Splitting fields

Normal extensions

Separable extensions

Simple extensions

Normal closures

Galois extensions

Automorphisms

Fixed fields

Fundamental theorem

Cyclic extensions

Solvable groups

Normal series

Extensions and subgroups

Insolvability of symmetric groups

Solution by radicals