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Math Help - Are these vector spaces isomorphs?

  1. #1
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    Are these vector spaces isomorphs?

    Let K1={a + (2^0.5)*b} | a,b rational numbers}, and K2={a + (3^0.5)*b} | a,b rational numbers} be two fields with the common multiplication and addition. Isomorphs are the following vector spaces :
    (Q^n ., +; K1) and (Q^n ., +; K2) ?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Are these vector spaces isomorphs?

    Quote Originally Posted by gotmejerry View Post
    Let K1={a + (2^0.5)*b} | a,b rational numbers}, and K2={a + (3^0.5)*b} | a,b rational numbers} be two fields with the common multiplication and addition. Isomorphs are the following vector spaces :
    (Q^n ., +; K1) and (Q^n ., +; K2) ?
    If you are asking whether \mathbb{Q}[\sqrt{2}]=\left\{a+b\sqrt{2}:a,b\in\mathbb{Q}\right\} and \mathbb{Q}[\sqrt{3}]=\left\{a+b\sqrt{3}:a,b\in\mathbb{Q}\right\} are isomorphic as vector spaces over \mathbb{Q}, then I have only one question--what are the dimensions of each of these spaces.
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