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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 subfields of $F$?

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

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    I am sorry,sir.It is just typo.I have edited my post.2012-12-19

2 Answers 2

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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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    We know that if a finite field$F$has characteristic p (prime), then F has cardinality pr where r=[F:Fp]. So r is a divisor of 12. I think proper subfield of $F$ is atmost 5.2014-03-29