How can I show that every finite field F has a characteristic of the field using its definition?
Thank you for any help.
I am sure you are familar with the rule that all we need to do is check whether,
$\displaystyle n\cdot 1=0$ for some positive integer to show there exists a charachteristic.
Now, the additive group of the field is a finite group because the field is finite. Thus, $\displaystyle 1\in <F,+>$ has finite order. Thus, $\displaystyle n\cdot 1 = 0$ where $\displaystyle 0$ is the identity element in $\displaystyle <F,+>$. Thus, it must have a non-zero charachteristic.