Why can't a field have exactly 6 elements? I tried to go over the axioms but couldn't figure out anything. Help please

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- Jan 23rd 2008, 11:08 PMnamelessguyQuestion related to field
Why can't a field have exactly 6 elements? I tried to go over the axioms but couldn't figure out anything. Help please

- Jan 23rd 2008, 11:34 PMJaneBennet
The number of elements in a finite field must be $\displaystyle p^k$, where

*p*is a prime and*k*is a positive integer. 6 is not of the form $\displaystyle p^k$. - Jan 24th 2008, 08:00 AMThePerfectHacker
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 $\displaystyle \phi (n) = n\cdot 1$.