# Question related to field

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• January 24th 2008, 12:08 AM
namelessguy
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
• January 24th 2008, 12:34 AM
JaneBennet
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 $p^k$.
• January 24th 2008, 09:00 AM
ThePerfectHacker
Quote:

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 $\phi (n) = n\cdot 1$.