Subspace question

  • Thread starter Diffy
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  • #1
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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?
 

Answers and Replies

  • #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.
 
  • #3
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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).
 
  • #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.
 
  • #5
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Thanks I wasn't thinking about it like that. I appreciate the insight.
 

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