# Thread: Field Characteristic

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.
I do not follow you.
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,
$n\cdot 1=0$ 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, $1\in $ has finite order. Thus, $n\cdot 1 = 0$ where $0$ is the identity element in $$. Thus, it must have a non-zero charachteristic.