Why can't a field have exactly 6 elements? I tried to go over the axioms but couldn't figure out anything. Help please
We can make it even easier than Jane Bennet said. It can be easily proven* that the characteristic of a field (in fact an integral domain) must be $\displaystyle 0$ or $\displaystyle p$ (a prime).
*)Define $\displaystyle \phi: \mathbb{Z}\mapsto F$ as $\displaystyle \phi (n) = n\cdot 1$.