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

Thank you for any help.

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- Jan 25th 2007, 01:59 PMPvtBillPilgrimField 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 PMThePerfectHacker
- Jan 25th 2007, 03:29 PMPvtBillPilgrim
I need to show the latter, that every finite field has a characteristic.

- Jan 25th 2007, 03:52 PMThePerfectHacker
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.