TITLE: Field Isomorphisms
QUESTION [6 upvotes]: Suppose $F/L$, $F'/L$, $L/K$ finite extensions of fields. If $F$, $F'$ isomorphic over $K$ then does it follow that they are isomorphic over $L$? I think probably not, but I can't come up with a counterexample. I've tried thinking about splitting fields of quartics or about function fields and neither has given me any joy. Obviously finite fields are no good. Could someone give me a hint? Thanks!

REPLY [4 votes]: Both $F = \Bbb{Q}(\sqrt[4]{2})$ and $F' = \Bbb{Q}(i\sqrt[4]{2})$ are isomorphic to $\Bbb{Q}[x]/(x^4 - 2)$, hence $\Bbb{Q}$-isomorphic, ($x^4 - 2$ is irreducible over $\Bbb{Q}$, so these are field extensions. The actual isomorphism is given by $\sqrt[4]{2} \mapsto i\sqrt[4]{2}$). However, they are not isomorphic over $L = \Bbb{Q}(\sqrt{2})$, as $x^2 + \sqrt{2} \in L[x]$ splits in $F'$, but not in $F$.