The field Q(i)={a+ib∣a,b∈Q} and Q(√2)={a+√2b∣a,b∈Q}.Then Show that
(a) Q(i) and Q(√2) are isomorphic as Q-vector spaces
(b) Show that Q(i) and Q(√2) are not isomorphic as fields
They are isomorphic as vector spaces because they have the same degree over $\displaystyle \mathbb{Q}$ so they are both isomorphic to $\displaystyle \mathbb{Q}^2$.
Hint: If $\displaystyle \phi: \mathbb{Q}(i) \to \mathbb{Q}(\sqrt{2})$ is an isomorphism then $\displaystyle \phi ( a ) = a \text{ for }a\in \mathbb{Q}$.(b) Show that Q(i) and Q(√2) are not isomorphic as fields
Try to get a contradiction from that.