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Math Help - field and vector spaces

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by makenqau View Post
    (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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