# Characteristic of infinite integral domain.

Give an example of infinite integral domain that has characteristic 3.

I know the result which states that finite integral domain has characteristic either 0 or prime.We have example of infinite integral domain that is $\Bbb Z$ but it has characteristic $0$.I am stuck here how to form an example of infinite integral domain with characteristic 3?

• Just as a side remark: a non-zero finite ring with $1$ without non-zero zero-divisors is always a field. And the fact that the characteristic is either $0$ or prime holds for all non-zero rings with $1$ without non-zero zero-divisors (commutative or not, finite or not). – 57Jimmy Apr 10 '18 at 8:21

You can consider the ring $R = \mathbb{F}_3[x]$ of univariate polynomials over the finite field of $3$ elements. As $\mathbb{F}_3$ is a field, thus an integral domain, $R$ will be an integral domain too, and it will have characteristic $3$. Clearly, $R$ is infinite.
• Yes,I got it.I can take $\Bbb Z_3[x]$ as a example. – ASHWINI SANKHE Apr 10 '18 at 7:51
• Indeed. By the way, this example works for any prime characteristic $p$ (just replace $\mathbb{Z}_3$ by $\mathbb{Z}_p$). – P. Senden Apr 10 '18 at 7:57