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**ThePerfectHacker** Let $\displaystyle k$ be a field, let $\displaystyle K=k(t)$ be the function field of $\displaystyle k$.

Let $\displaystyle \alpha \in K$ and set $\displaystyle \alpha = \tfrac{f(t)}{g(t)}$ in reduced terms, with $\displaystyle \alpha \not \in k$.

Now construct $\displaystyle F = k(\alpha)$.

Note $\displaystyle t\in k(t)$ solves the polynomial $\displaystyle \alpha f(x) - g(x)\in F[x]$.

This is a non-zero polynomial because otherwise it would contradict $\displaystyle \alpha \not \in k$.

How do we prove this polynomial is irreducible over $\displaystyle F$?