TITLE: Forall exists formula
QUESTION [6 upvotes]: This is a reference request.
Let $A$ be a formula in the language of rings which is of the form $\forall_{x_1}\dots\forall_{x_n}\exists_{y_1}\dots\exists_{y_m} F$, where $F$ is quantifier-free. I once read that if $A$ is valid for all finite fields, then it is valid for $\mathbb C$. Where would I find a proof of this statement?

REPLY [11 votes]: I don't know a reference but it's not that hard to prove.
Let's call your formula $\varphi$. Then by classical arguments, the models of $\varphi$ are closed under directed union, in particular since $\varphi$ holds in any finite field, it holds in any $\overline{\mathbb{F}_p} = \displaystyle\bigcup_{n<\omega}\mathbb{F}_{p^n}$, the algebraic closure of the field with $p$ elements, which is the directed union of its finite subfields. 
But then by Los's theorem, $\varphi$ holds in $\displaystyle\prod_{p\in \mathbb{P}}\overline{\mathbb{F}_p}/\mathcal{U}$ for any $\mathcal{U}$ ultrafilter on $\mathbb{P}$. Picking a non-principal ultrafilter yields a field isomorphic to $\mathbb{C}$, so the formula holds in $\mathbb{C}$ too.