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For $d\in\mathbb{Q}$, how could one show that $Q(\sqrt{d})$ lies in a cyclotomic extension of $\mathbb{Q}$ without using the Kronecker-Weber theorem?

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1) Use the decomposition of the discriminant into prime discriminants, and show that the corresponding quadratic fields are cyclotomic using the fact that in these fields, only one prime is ramified. –  franz lemmermeyer Apr 18 '12 at 7:42
    
2) Use quadratic Gauss sums. –  franz lemmermeyer Apr 18 '12 at 7:42
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@rayjsh: It appears you tried to edit your own question, but were not logged in-the edit comes from an anonymous user. If you log in, I think you should be able to edit it. I wouldn't want to put those words in your mouth if they aren't yours. –  Ross Millikan Apr 18 '12 at 13:07

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