Best Answer:
Well, if it's characteristic zero, it's characteristic zero... that's one case. So now, assume it is not -- that there is a solution, so that:

1 + 1 + .... + 1 = 0

where there are n of the 1s. We will let n be the minimal possible number for this -- that means char = n. If n is composite, say n=km (for n>k,m>1). Then:

(1+1+...+1)(1+1+...+1) = 0

we have factored this into a sum of k 1s and a sum of m 1s.

But this is an integral domain, so either

1+1+...+1 = 0 [k 1s]

or

1+1+...+1 = 0 [m 1s]

But we said n was the minimal such value, and since either k<n or m<n provides a contradiction, we know no such k & m pair exists.

What does that mean? It means n has no factors. In other words, it is prime.

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