Let’s say we have a finite field extension $L/K$. Suppose we have that $L(\alpha )$ and $L(\beta )$ are isomorphic as fields (finite extensions) with an isomorphism $\phi $.
Now viewing $L(\alpha ) $ and $L(\beta )$ as $L$-vector spaces. Are these vector spaces necessarily isomorphic?
I would have thought so and would have thought $\phi $ is a linear isomorphism. But for that to be the case, for all $a \in L$ and $x \in L(\alpha )$ we would need $\phi (ax)= a\phi (x) $ but the field isomorphism only means that $\phi (ax)=\phi(a) \phi(x) $.
Any clues on to what is going on here?