# Integral extension of fields is a finite vector space extension

I want to show that if $E$ and $F$ are fields, and $E$ is an integral extension of $F$, then $E$ as a vector space over $F$ has finite dimension. Does anyone have a proof of this, or can reference one?

Thanks.

• This is basically Hilbert's Nullstellensatz. – Lord Shark the Unknown Mar 16 '18 at 18:30
• I am trying to prove this in order to prove the Weak Nullstellensatz – Daven Mar 16 '18 at 18:30
• See this link, under "Corollary, Weak Nullstellensatz": dpmms.cam.ac.uk/~sjw47/Lecture16-18.pdf . It concludes with "A is integral over k as required", but that is not exactly what it set out to show (that the extension is finite). I am trying to equate these two statements. – Daven Mar 16 '18 at 18:32
• @Daven: You might find this useful. – Prasun Biswas Mar 16 '18 at 19:09
• It seems to me that an algebraic closure of $\mathbb Q$ is integral over $\mathbb Q$, and that it is infinite dimensional as a $\mathbb Q$-vector space. – Pierre-Yves Gaillard Mar 16 '18 at 19:20