field and vector spaces

**makenqau** · December 2nd 2008, 09:08 AM

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

**ThePerfectHacker** · December 2nd 2008, 09:58 AM

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.

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