Thread: 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.

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?

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

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,
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, has finite order. Thus, where is the identity element in . Thus, it must have a non-zero charachteristic.