Field Characteristic

• Jan 25th 2007, 01:59 PM
PvtBillPilgrim
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.
• Jan 25th 2007, 02:20 PM
ThePerfectHacker
Quote:

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?
• Jan 25th 2007, 03:29 PM
PvtBillPilgrim
I need to show the latter, that every finite field has a characteristic.
• Jan 25th 2007, 03:52 PM
ThePerfectHacker
Quote:

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.