= Solution
Fix $n\geq1$ and let $\mathbb E_i/\mathbb F_i$ be the unique degree-$n$ finite-field extension. <Łoś theorem> shows that
$$
\mathbb E=\prod_i\mathbb E_i/\mathcal U_2
$$
is a field extension of $\mathcal F$ of degree $n$: ultraproducts of chosen bases satisfy the first-order linear-independence and spanning statements.
Conversely, let $\mathcal F(\alpha)/\mathcal F$ have degree $n$, with irreducible minimal polynomial $f$. Represent its coefficients by polynomials $f_i$. Irreducibility in fixed degree is first-order, so $f_i$ is irreducible of degree $n$ for $\mathcal U_2$-almost every $i$. Its root generates $\mathbb E_i$, and the ultraproduct of these roots induces an $\mathcal F$-isomorphism $\mathcal F(\alpha)\cong\mathbb E$. Hence the degree-$n$ algebraic extension exists and is unique.
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