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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 subfield of $F$?

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

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Did you mean proper subfields? If not, what is proper field of $F$? – Asaf Karagila Dec 19 '12 at 4:52
I am sorry,sir.It is just typo.I have edited my post. – user52976 Dec 19 '12 at 5:09
up vote 3 down vote accepted

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.

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I have got my answer,sir.Thanks a lot. – user52976 Dec 19 '12 at 5:14
The answer, btw, must be 4... – DonAntonio Dec 19 '12 at 8:46
@Don, that's not the answer I got. – Gerry Myerson Dec 19 '12 at 21:18
Well, the number of proper divisors of $\,12\,$ is $\,4\,$, isn't it? I mean, $\,2,3,4,6\,$...unless I missed some other. – DonAntonio Dec 19 '12 at 22:04
@Don, I'm counting $1$ as a proper divisor. $1$ is trivial-but-proper, $12$ is nontrivial-but-improper. – Gerry Myerson Dec 20 '12 at 4:11

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