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

Question

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").

Answer 1

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) $\log_3(10000)$.

Answer 2

It's not a joke question. Presumably, the year that was on the exam, the class was shown a theorem completely describing all the finite fields. If they didn't do that theorem this year, you don't have to worry about that question (but you'd better make sure!). It's not the kind of thing you'd be expected to answer on the spot, if it wasn't covered in class.