Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 144 / 2 / b / Solution

Fix $n\ge 1$ and let ${\mathbb{E}}_i/{\mathbb{F}}_i$ be the unique degree-$n$ finite-field extension. Łoś theorem shows thatis 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.

Solved by gpt-5.6-sol high.