# Finite extensions of characteristic zero fields are simple.

I have been recently been introduced to field theory, and been told I should be able to understand a proof for this the result that: every finite extension of a field of characteristic zero is a simple extension.

I have not been directed to a specific proof, just told that I should be able to understand it. However, the proof given in my textbook uses results and concepts I have not yet been exposed to.

Is there an elementary proof given of this result that I may be able to understand with only a superficial understanding of the field extensions, algebraic extensions, and some idea of what a separable extension is?

• Well in fields of characteristic $0$ every irreducible polynomial is separable. Then you need to know some stuff like separability degree is equal to extension degree iff the extension is separable for finite extension and some stuff like this... – user2345215 Mar 6 '14 at 1:48
• "Finite separable extensions are simple" is the primitive element theorem: en.wikipedia.org/wiki/Primitive_element_theorem – Ian Coley Mar 6 '14 at 2:03
• Are there any proofs of the primitive element theorem that would be accessible to someone with my limited background? I have tried looking for something, but everything goes over my head since I don't know the associated terminology. I'd appreciate if anyone could perhaps outline the proof in language that I might understand (if something like this could could be condensed to a short explanation). – numberman Mar 6 '14 at 2:09