Finding the total number of proper subfields of $F$?

Question

I was thinking about the following problem:

Let $F$ be a field with $5^{12}$ elements.Then how can I find the total number of proper subfields of $F$?

Can someone point me in the right direction? Thanks in advance for your time.

Answer 1

I take it you mean, proper subfield.

Can you show that any subfield of $F$ contains the field of $5$ elements?

Can you show that any subfield must contain $5^r$ elements, for some $r$?

Can you show that the degree of such a subfield (over the field of $5$ elements) must be $r$? and must be a divisor of the degree of the field of $5^{12}$ elements?

Can you show that a finite field has at most one subfield of any given number of elements?

If you can do all those, you have your answer.