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Math Help - integral domains, fermats

  1. #1
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    integral domains, fermats

    a) Show that the characteristic of an integral domain D must be either 0 or prime.
    b) Show that 1 and p-1 are the only elements of the field Zp that are their own multiplicative inverse.
    c) Use Fermat's theorem to show that for any positive integer n, the integer n^(37) - n is divisible by 383838.

    (:
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  2. #2
    Junior Member Dark Sun's Avatar
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    Hello littlebbygurl, I have the proof of a) for you:

    If 1 has infinite order under addition, then the characteristic of the integral domain is 0.

    Suppose that 1 has order n, and n=st, where 1\leq s,t\leq n. Then, we have:

    0=n\cdot 1=(st)\cdot 1=(s\cdot 1)(t\cdot 1)(*)

    Therefore, s\cdot 1=0 or t\cdot 1=0. However, since n is the least positive integer s.t. n\cdot=0, it must be so that s=n or t=n, proving that n is prime.

    If (*) is not trivial to you, you may want to prove it as well. The same is true of line one of the proof.

    Hope this helps, happy mathing!
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