#### 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

Math 403

#### 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