## Tag Archives: finite extension

### Simple Extension

$\mathbf{Defination:}$ Let $E/F$ be a finte extension, if there is $u\in E$ such that $E=F(u)$, then we call $u$ is a primitive element of the extension $E/F$. And this extension is called simple exentsion.

It is very important to know when a extension is a simple extension.The following theorem Emil Artin

$\mathbf{Thm:(Primitive\, Element\, Thm)}$ $E/F$ is a simple extension if and only if there are finitely many intermediate field $\displaystyle K$ between $E$ and $F$.
We are not interested in the proof of this theorem. But in the consequence of this theorem.

$\mathbf{Corollary:}$ Let $E/F$ be a finite separable extension, then $E/F$ is a simple extension. Especially when $\displaystyle char(F)$ is 0, all finite extension is a simple one.