The characteristic of the field $GF(p^n)$

Question

How to show that characteristic of the field $GF(p^n)$ is $p$?

I have come across this fact on Wikipedia webpage, but don't know how to prove it.

Thanks

Answer 1

Let $r \in \mathbb{N}$ be the characteristic of $GF(p^n)$ .

We know that the characteristic of a domain is always $0$ or a prime number. In the case of a finite field, it can't be $0$ because otherwise the field would contain a copy of $\mathbb{Z}$. So the characteristic of $GF(p^n)$ is a prime.

Then we have $r \cdot 1 = 0 $, but a field is in particular an abelian group with respect to $+$ and so $$r \mid p^n \Rightarrow r = p$$