Let $\mathscr{L}=\{+,·\}$ be the language of the theory of fields. Let $\phi$ be a sentence in this language. Show, using the compactness theorem of first-order logic, that if $\phi$ holds in finite fields of arbitrarily large characteristic, then $\phi$ holds in a field of characteristic $0$.
So How would I use compactness for a problem like this? It seems like the generalization metatheorem would be included, or something similar.
Consider a set $\Gamma$ of formulas consisting of $\phi$ and $i \neq j$, for $i,j \in \{1,2,3,\ldots\}$ (that is, of course, a meta description of the formulas). If $\phi$ holds in finite fields of any characteristic, then every finite subset of $\Gamma$ has a model and by compactness $\Gamma$ has a model $M$. But then $M$ must be infinite (and with characteristic 0 because $i \neq 0$ for $i=1,2,\ldots$) and $M \vDash \phi$