Counting Subfields in F_p: Algebraic Result?

[Diffy]

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?

 

[morphism]

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.

 

[Diffy]

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).

 

[HallsofIvy]

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.

 

[Diffy]

Thanks I wasn't thinking about it like that. I appreciate the insight.