# Math Help - field and vector spaces

1. ## field and vector spaces

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

2. Originally Posted by makenqau
(a) Q(i) and Q(√2) are isomorphic as Q-vector spaces
They are isomorphic as vector spaces because they have the same degree over $\mathbb{Q}$ so they are both isomorphic to $\mathbb{Q}^2$.

(b) Show that Q(i) and Q(√2) are not isomorphic as fields
Hint: If $\phi: \mathbb{Q}(i) \to \mathbb{Q}(\sqrt{2})$ is an isomorphism then $\phi ( a ) = a \text{ for }a\in \mathbb{Q}$.
Try to get a contradiction from that.