If $F$ is a finite field then is it necessary that $|F|=p^n$ for some prime $p$ and positive integer $n$? I know that given prime $p$ and positive integer $n$, there is a field such that $|F|=p^n$? But is the converse always true?
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Yes, it is true. If $F$ is a finite field, then its characteristic is finite, so is a prime number $p$. This implies that $F$ is a vector space over $k=\mathbb{F}_p$. Denoting by $n$ the dimension of this vector field, the field $F$ has $p^n$ elements.
Jérémy Blanc
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vector space, not vector field. :-) – Adam Hughes Jan 07 '15 at 00:10
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Thanks. Yes, it is a field which is a vector space over a field, but not a vector field... too many fields :-) – Jérémy Blanc Jan 07 '15 at 00:11