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My study group thinks this is false since we couldn't come up with any.

user70710
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    See http://math.stackexchange.com/questions/42143/can-you-construct-a-field-with-4-elements ... edit: also http://en.wikipedia.org/wiki/Field_(mathematics)#Second_example:_a_field_with_four_elements – Neal Apr 03 '13 at 04:32
  • There is a field with $p^e$ elements for every prime $p$ and every $e\in \mathbb{N}$. – Alexander Gruber Apr 03 '13 at 05:13

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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.

stoogebag
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