Are simple extensions of isomorphic fields isomorphic?

Question

If $F(u) \cong F(v)$, and both are subfields of a larger extension, $K$, then for $k \in K$, is $F(u)(k) \cong F(v)(k)$?

Answer 1

Nope.. For a very concrete example you can simply pick nonalgebraic extensions, say in $\mathbb{Q} \subset \mathbb{C}$. So let $F = \mathbb{Q}$, $u = \pi$ and $v = e$. Since both $u$ and $e$ are transcedent over $\mathbb{Q}$, the fields $\mathbb{Q}(u)$ and $\mathbb{Q}(v)$ are isomorphic, but as noted in the comments we can now pick $k = \pi$ for a clear counterexample.