# 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

## 2 Answers

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) $\log_3(10000)$.

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OK, so now heishe knows there is a field of order 9, but heishe doesn't know what it is, doesn't know how to show that it's isomorphic to every other field of order 9, and doesn't know how to show there aren't any fields of order 6. I'm glad heishe is happy with the answer, but it's a very shallow sort of knowledge that has been transmitted here. –  Gerry Myerson Jul 8 '11 at 2:01
You're right in a sense; my answer doesn't explain the technology of the result and the same information could have been found by an (informed) Google search. On the other hand, it does answer the question. While some people go to great lengths to write clear and comprehensive answers (something which I can only admire), I do not feel it should be a requirement to write up a derivation of any and all results one needs in answering a particular question. In any case, if, as you presume, heishe hasn't seen this theorem in class, it doesn't really matter much. –  Miha Habič Jul 8 '11 at 20:53
You are right, in all senses. I wouldn't expect anyone to write up, in this forum, a derivation of all the results you give in your answer, especially as those derivations are widely available elsewhere. I think I was mostly letting off steam, and maybe trying to point heishe to some further directions to explore when the exam is over. –  Gerry Myerson Jul 9 '11 at 5:22

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.

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