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Math Help - P-adic integers and roots of unity

  1. #1
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    Unhappy P-adic integers and roots of unity

    Show that Zp contains all the (p-1)th roots of unity.
    For which primes p does Zp contains primitive fourth roots of unity.
    Here Zp is the set of p-adic integers.
    Proving that it has a (p-1)th root of unity is easy, but ALL roots is another matter. Please help me with these questions..
    I think for the second question, p has to be 5, but maybe there are other answers that i didn't think of.
    Last edited by lahuxixi; November 21st 2013 at 05:47 PM.
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  2. #2
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    Re: P-adic integers and roots of unity

    This is what i have done
    Let f(x)=x^(n-1)-1
    there exist x such that gcd(x,p)=1, just let x=1 and x^(n-1)=1 mod p.
    f'(1)=/=0 since it's still 1. so there exist a solution in Zp that f(x)=0
    And part 2, since with part 1, Zp has all (p-1)th root, it's obvious that if p=5, Zp had all 4th roots, one of them is a primitive root. But that answer is too stupid
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  3. #3
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    Re: P-adic integers and roots of unity

    solved
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