A question concerning non-algebraic extension.

Question

Let $\tau:F \to \overline{F}$ be a field embedding.

Then is $\overline{F}/\tau(F)$ algebraic extension?

I don't think so but I cannot find a counterexample.

Would you let me know a counterexample?

Answer 1

It is a bit unclear what is asked, but judging from the comments of others the following interesting interpretation comes to mind. We start with a field $F$ and its algebraic closure $\overline{F}$, and then we wonder whether $\overline{F}/\tau(F)$ is algebraic for all field homomorphisms $\tau:F\to\overline{F}$.

The answer to that question is 'No'.

Let $F=\Bbb{Q}(x_0,x_1,\ldots)$ be a purely transcendental extension of the rationals of a countably infinite transcendence degree. Let us define $\tau:F\to\overline{F}$ by declaring $\tau(x_i)=x_{i+1}$ and extending that to a homomorphism of fields in the obvious way. Then the element $x_0\in\overline{F}$ will be transcendental over $\tau(F)$.