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Math Help - P-adic Woes

  1. #1
    diffyq
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    P-adic Woes

    I've been working on this problem for my analysis class for a while now, and I don't have any idea where to go. Any help at all would be greatly appreciated. Thanks!

    Suppose p is an odd prime. Define the set P = {x in Q_p: |x|_p < 1}, that is, the p-adic numbers that have p-adic absolute value less than 1. Define U_1 = P + 1. Prove that the function f(x) = x^2 from U_1 to U_1 is a bijection.
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  2. #2
    Senior Member
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    You need to show that the binomial expansion of (1+x)^{1/2} is p-adically convergent for |x| < 1. It's sufficient to prove the terms tend to zero: the r-th term is (1/2)(-1/2)(...)(1/2-r) x^r / r! = (+-) (2r)! x^r / 2^r (r!)^2. Since p is odd this is an integer (binomial coefficient) times x^r and so goes to zero.
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