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Order of element

  1. Aug 19, 2013 #1
    Is there a term for the least k such that a^k = 1 in some ring R (if it exists, specifically in a finite ring)? Is there a way to get an upper bound in general? I know that it's hard to get even in say F_p, but I'm just looking for some conditions that would help me understand what it looks like.
     
  2. jcsd
  3. Aug 19, 2013 #2

    micromass

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    The term order should work well.

    Let ##n## be the number of invertible elements in the ring, then ##n## is an upper bound.

    Well, you know that the invertible elements in ##\mathbb{F}_p## form a cyclic group of order ##p-1##. So finding the orders of the elements in equivalent to finding the orders of elements in ##\mathbb{Z}_{p-1}##, which is easy. So you know exactly which orders show up and which don't. The hard part is finding an explicit isomorphism between ##\mathbb{Z}_{p-1}## and the invertible elements in ##\mathbb{F}_p## (or equivalently: finding an element in ##\mathbb{F}_p## with order ##p-1##).
     
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