My study group thinks this is false since we couldn't come up with any.
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2$\begingroup$ See math.stackexchange.com/questions/42143/… ... edit: also en.wikipedia.org/wiki/… $\endgroup$– NealCommented Apr 3, 2013 at 4:32
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$\begingroup$ There is a field with $p^e$ elements for every prime $p$ and every $e\in \mathbb{N}$. $\endgroup$– Alexander Gruber ♦Commented Apr 3, 2013 at 5:13
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Consider the binary field $\mathbb{Z}_2$, and 'extend' it by adding a root of an irreducible polynomial (say, $x^2+x+1$) in the same way that you would 'extend' the real numbers to the complex numbers by adding a root of $x^2+1$.
This gives the field $GF(4)$, which should be easy enough to find information about.
