# Number of fields with characteristic of 3 and less than 10000 elements?

it's exam time again over here and I'm currently doing some last preparations for our math exam that is up in two weeks. I previously thought that I was prepared quite well since I've gone through a load of old exams and managed to solve them correctly.

However, I've just found a strange question and I'm completely clueless on how to solve it:

How many finite fields with a charateristic of 3 and less than 10000 elements are there?

I can only think of Z3 (rather trivial), but I'm completely clueless on how to determine the others (that is of course - if this question isn't some kind of joke question and the answer really is "1").

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There is (up to isomorphism) one field with $3$ elements, one with $9$, one with $27$, one with $81$. Continue. Only four more to go. – André Nicolas Jul 7 '11 at 13:57

Galois' theorem states that there exists precisely one finite field of characteristic $p$ with $p^n$ elements for each $n$ and that these are all of the finite fields of characteristic $p$. So all you have to do is calculate (or estimate) $\log_3(10000)$.