Solution of x^p = 1 in a finite field of char p

I was wondering if x^p = 1 implies that x =1 in a finite field of characteristic p, or in other words
If x^p = 1 (mod p), does that imply that x = 1 (mod p) ?

If so why?

Hello,

Quote:

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Originally Posted by petersmith

I was wondering if x^p = 1 implies that x =1 in a finite field of characteristic p, or in other words
If x^p = 1 (mod p), does that imply that x = 1 (mod p) ?

If so why?

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x cannot be =0, otherwise, it wouldn't be possible that x^p=1
Considering that x is an element of the field, , we have by Fermat's little theorem :

So if , then ...

Hmmm I think it killed the question... or I've made a mistake ^^'

If then .
Thus if .

Quote:

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Originally Posted by Moo

Hmmm I think it killed the question... or I've made a mistake ^^'

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What you did is correct if .
However, a finite field need not be a the integers modulo .
Thus, it does not cover all the cases.

This is Mine 1:)4:):)th Post!!!