What is an example of transcendental extension such that a monomorphism cannot be extended?

Question

Let $E/F$ be a field extension and $\alpha\in E$ be transcendental over $F$.

Let $\bar F$ be the algebraic closure of $F$ and $\sigma:F\rightarrow \bar{F}$ be a field monomorphism.

What is an example of $\alpha$ such that $\sigma$ cannot be extended to a field monomorphism $\tau:F(\alpha)\rightarrow \bar{F}$?

Answer 1

Any example will do, such as $F=\mathbb Q$, $\alpha=\pi$. Since $\overline F$ is algebraic over $F$, $\tau(\alpha)$ would have to be algebraic, say $f(\tau(\alpha))=0$ with $f\in F[X]$. Since $f(\alpha)\ne 0$, $\tau$ fails to be monomorphic.

Answer 2

Take $\sigma=id: F=\mathbb C\to \bar F=\mathbb C$ and choose $E$ and $\alpha$ any way you like: $\sigma$ won't extend to $\mathbb C(\alpha)$.