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Math Help - Characteristic of an integral domain

  1. #1
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    Characteristic of an integral domain

    Let A be a finite integral domain. Prove that if there is a nonzero a in A such that 256 * a = 0, then A has characteristic 2.

    I'm not at all sure how to do this. Any advice is appreciated. Thanks for your time!
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Characteristic of an integral domain

    First we know that the characteristic of an ID is p, where p is prime (since ID is finite) .

    Second notice that the additive order of every non zero element in the integral domain is equal to characteristic p.
    Why?

    Take  x \in D such that  x \not = 0 . Let the additive order of x be n. Then since  x \not = 0 , n > 1. Since the characteristic is p, we know that  px = 0 . Thus this means that n must divide p. Since the only things that divide p are 1 and p, since we said n > 1, it must be that n = p .

    This means that p must divide 256 = 2^8.

    Since 2 is the only prime which divides 256. So A has char 2.
    Last edited by jakncoke; December 1st 2012 at 07:47 PM.
    Thanks from jzellt
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