I came across an exercise, in which I had to prove, that $\mathbb{Q}(i)$ and $\mathbb{Q}(\sqrt{2})$ are isomorphic as vector spaces but not as fields. This is quite an easy exercise, as $-1$ can be represented as $i^2$ over $\mathbb{Q}(i)$ but not as $x^2$ with $x \in \mathbb{Q}(\sqrt{2})$.
However I was wondering, if the statement "two field extensions are isomorphic as fields implies field extensions are isomorphic as vector spaces" is true.
Best Answer
A field isomorphism of field extensions is necessarily a linear isomorphism considering the field extensions as vector spaces. The definition of a field isomorphism is more restrictive than just a linear isomorphism.