In my Galois theory class I've seen the next theorem
Given $F:K$ and $F':K'$ two field extensions, $u\in F$ and $u'\in F'$ two elemets of the bigger fields and $\sigma : K \rightarrow K'$ a field isomorphism. Then, if $u$ and $u'$ are trascendent over $F$ and $F'$ there exists a field isomorphism $\tilde{\sigma}:F\rightarrow F'$ that extends $\sigma$ to the bigger fields.
I'm new to field extensions, however, I belive this is not necesarily true unless $F=K(u)$ and $F'=K'(u')$.
In this case, as $K(u)\cong K(x) \cong K'(x) \cong K'(u')$ where the middle equivalence holds because there exists $\sigma $ an isomorphism from $K$ to $K'$.
However, back to the original theorem, I don't know how to prove it with my current knowledge and I'm thinking it might be a mistake as in a complementary handbook we were given the same theorem is written with $F,F'=K(u), K'(u')$.
I'd appreciate if anyone could tell me wether the original theorem is true and in case it is could give me some hints on how to solve it. In the case the therorem only holds for the specific case i prooved, could anyone make sure my proof is right?
Thanks for the help and sorry for my poor english.