# Thread: Characteristic of an integral domain

1. ## Characteristic of an integral domain

Let A be a finite integral domain. Prove that if there is a nonzero a in A such that 256 * a = 0, then A has characteristic 2.

I'm not at all sure how to do this. Any advice is appreciated. Thanks for your time!

2. ## Re: Characteristic of an integral domain

First we know that the characteristic of an ID is p, where p is prime (since ID is finite) .

Second notice that the additive order of every non zero element in the integral domain is equal to characteristic p.
Why?

Take $\displaystyle x \in D$ such that $\displaystyle x \not = 0$ . Let the additive order of x be n. Then since $\displaystyle x \not = 0$, n > 1. Since the characteristic is p, we know that $\displaystyle px = 0$. Thus this means that n must divide p. Since the only things that divide p are 1 and p, since we said n > 1, it must be that n = p .

This means that p must divide 256 = 2^8.

Since 2 is the only prime which divides 256. So A has char 2.