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Math Help - Question related to field

  1. #1
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    Question related to field

    Why can't a field have exactly 6 elements? I tried to go over the axioms but couldn't figure out anything. Help please
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  2. #2
    Senior Member JaneBennet's Avatar
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    The number of elements in a finite field must be p^k, where p is a prime and k is a positive integer. 6 is not of the form p^k.
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  3. #3
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    Quote Originally Posted by namelessguy View Post
    Why can't a field have exactly 6 elements? I tried to go over the axioms but couldn't figure out anything. Help please
    We can make it even easier than Jane Bennet said. It can be easily proven* that the characteristic of a field (in fact an integral domain) must be 0 or p (a prime).



    *)Define \phi: \mathbb{Z}\mapsto F as \phi (n) = n\cdot 1.
    Last edited by ThePerfectHacker; January 24th 2008 at 09:12 AM.
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