Let $\mathcal{L}$ be the first-order language of theory of rings. Let A,B be two infinite fields. Suppose that A is existentially closed in B. We know that
A is existentially closed in B iff B can be embedded in a $\mathcal{L}$-structure B* which is elementary extension of A
Since A is existentially closed in B, A is also algebraically closed in B. The above proposition also tell us that A is algebraically closed in B* since B*/A is a regular extension. Here is my question :
Is B necessary algebraically closed in B* ?
I guess $\mathbb{C}$ and the rational function field $\mathbb{C}(T)$ may be a counterexample.