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Math Help - ring of fractions

  1. #1
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    ring of fractions

    Consider \mathbb{Z}[2\sqrt2]=\{a+2b\sqrt{2}|a,b \in \mathbb{Z}\}, is it's ring of fractions \{c+2d\sqrt{2}|c,d \in \mathbb{Q}\}?
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    Consider \mathbb{Z}[2\sqrt2]=\{a+2b\sqrt{2}|a,b \in \mathbb{Z}\}, is it's ring of fractions \{c+2d\sqrt{2}|c,d \in \mathbb{Q}\}?
    You need to show that if F is a field that contains \mathbb{Z}[2\sqrt{2}] then F contains \{ c+2d\sqrt{2}|c,d\in \mathbb{Q}\}.

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  3. #3
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    I was wondering for \mathbb{Z}[2\sqrt{2}], is it's ring of fractions \{ c+2d\sqrt{2}|c,d\in \mathbb{Q}\} or \{\frac{a+2b\sqrt2}{c+2d\sqrt2}|a,b,c,d\in\mathbb{  Z}\}?
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  4. #4
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    Quote Originally Posted by dori1123 View Post
    I was wondering for \mathbb{Z}[2\sqrt{2}], is it's ring of fractions \{ c+2d\sqrt{2}|c,d\in \mathbb{Q}\} or \{\frac{a+2b\sqrt2}{c+2d\sqrt2}|a,b,c,d\in\mathbb{  Z}\}?
    I think it is the second one because it contains all elements of the form \alpha/\beta where \alpha,\beta \in D^{\times}.
    Also, it is not really called "ring of fractions" but "field of fractions".
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