1. ## Field Characteristic

How can I show that every finite field F has a characteristic of the field using its definition?

Thank you for any help.

2. Originally Posted by PvtBillPilgrim
How can I show that every finite field F has a characteristic of the field using its definition?

Thank you for any help.
How can you show what?
That every finite field has prime chatachteristic?

Or are you asking to show that evert finite field has a charachteristic?

3. I need to show the latter, that every finite field has a characteristic.

4. Originally Posted by PvtBillPilgrim
I need to show the latter, that every finite field has a characteristic.
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.