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.