Simple extension

Description

A simple extension is an extension obtained from adjunction with only one element. I don’t think it has a relation ship to Einfache Gruppe.

Definition

Eine Field extension $L|K$ wird einfach genannt, wenn ein Element $\alpha \in L$ mit $L=K(\alpha )$ existiert. In diesem Fall nennt man $\alpha$ ein primitives Element der Erweiterung.

Properties

Vector space structure

It is known that field extensions have the structure of a vector field. In a simple extensions $K(\alpha )\mid K$ the degree of the Minimal polynomial ${\mu }_{\alpha ,K}$ is equal to the Degree (field extension) and the basis is given by $1,\alpha ,{\alpha }^2,...,{\alpha }^{n-1}$.

Finitely many intermediate fields

An extension $L\mid K$ has finitely many intermediate fields if and only if it is simple.

Examples

Satz vom primitiven Element

Jede endliche, seperable Erweiterung $L|K$ ist einfach.