Prove that adding one to itself is always different from 0

Hi,

**problem: **

.

If and are in , let be the least positive remainder obtained by dividing the (ordinary) sum of and by ,

and similarly, let be the least positive remainder obtained by dividing the (ordinary) product of and by

Prove that if is a field, then either the result of repeatedly adding to itself is always different from ,

or else the first time that it is equal to occurs when the number of summands is a prime.

**attempt:**

I look at where is the number of times 1 is added to itself.

, is prime since is a field.

when is a multiple of , .

For , .

For ,

I am really bad at this so any suggestion is very welcome!

Thanks