# Math Help - Question related to field

1. ## Question 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

2. The number of elements in a finite field must be $p^k$, where p is a prime and k is a positive integer. 6 is not of the form $p^k$.

3. Originally Posted by namelessguy
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 $0$ or $p$ (a prime).

*)Define $\phi: \mathbb{Z}\mapsto F$ as $\phi (n) = n\cdot 1$.