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Subspace question

  1. Oct 6, 2008 #1
    Consider the prime field F_p p a prime.
    How can I count the number of subfields there are?

    Is this a known result of algebra?
     
  2. jcsd
  3. Oct 6, 2008 #2

    morphism

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    This is actually very easy. Try to convince yourself that F_p can't have any proper subfields.

    In some sense F_p is the 'smallest' field of characteristic p.
     
  4. Oct 6, 2008 #3
    This was my gut, I could not conceive how you could have a subfield. So the answer is 2, (because I still want to count trivial subfields).
     
  5. Oct 6, 2008 #4

    HallsofIvy

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    The "known result" is actually a result from group theory: Every field is a group considering only its addition operation and every subfield is a subgroup of that group. The order of a subgroup must divide the order of the group.
     
  6. Oct 6, 2008 #5
    Thanks I wasn't thinking about it like that. I appreciate the insight.
     
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