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Math Help - Hilbert class field

  1. #1
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    Hilbert class field

    For the number fields K whose ring of integer is UFD,
    why the Hilbert class field of K is K itself???



    thanks
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  2. #2
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    Quote Originally Posted by chipai View Post
    For the number fields K whose ring of integer is UFD,
    why the Hilbert class field of K is K itself???

    thanks
    it's actually an if and only if statement. the reason is quite clear: \mathcal{O}_K, the ring of integers of K, is a UFD if and only if it's a PID because \mathcal{O}_K is always a Dedekind domain.

    on the other hand by definition H(K), the ideal class group of K, is trivial if and only if every ideal of \mathcal{O}_K is principal, i.e. if and only if \mathcal{O}_K is a PID. so |H(K)|, the class

    number of K, is 1 if and only if \mathcal{O}_K is a PID. finally the dimension of the Hilbert class field of K, as a vector space over K, is exactly |H(K)| and the result follows.
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