Let be a finte extension, if there is such that , then we call is a *primitive element *of the extension . 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

is a simple extension if and only if there are finitely many intermediate field between and .

We are not interested in the proof of this theorem. But in the consequence of this theorem.

Let be a finite separable extension, then is a simple extension. Especially when is 0, all finite extension is a simple one.