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Math Help - proving an isomorphism

  1. #1
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    proving an isomorphism

    Hi, could you please help me with the following question.


    How do i prove that the set of all complex numbers of the form a + bi with a, b in Q (rationals) is isomorphic to the field of fractions of the ring Z[i] of Gaussian integers?

    Thanks for your help.
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  2. #2
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    Quote Originally Posted by Louise View Post
    Hi, could you please help me with the following question.


    How do i prove that the set of all complex numbers of the form a + bi with a, b in Q (rationals) is isomorphic to the field of fractions of the ring Z[i] of Gaussian integers?

    Thanks for your help.
    looking at them as subsets of \mathbb{C}, they're more than just isomorphic. they are basically equal. because an element of the field of fractions of \mathbb{Z}[i] is in the form x=\frac{a+bi}{c+di}, where a,b,c,d \in \mathbb{Z}

    and cd \neq 0. but we have x=\frac{(a+bi)(c-di)}{c^2+d^2}=\frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i \in \mathbb{Q}(i). on the other hand if y \in \mathbb{Q}(i), then y=p+qi, for some p=\frac{a}{b}, \ q=\frac{c}{d}, with a,b,c,d \in \mathbb{Z}, and bd \neq 0. then

    y=\frac{ad + bc i}{bd}. clearly both ad + bc i and bd are in \mathbb{Z}[i] and thus y is in the field of fractions of \mathbb{Z}[i] because bd \neq 0.
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