# Integral Domain

• December 2nd 2008, 11:03 AM
ynn6871
Integral Domain
I am having troubles with the following problem:

Show that there is an integral domain with exactly 4 elements.

Any suggestion?
• December 2nd 2008, 11:54 AM
ThePerfectHacker
Quote:

Originally Posted by ynn6871
I am having troubles with the following problem:

Show that there is an integral domain with exactly 4 elements.

Any suggestion?

Let $F = \mathbb{Z}_2[x]$ - the polynomials with coefficients in $\mathbb{Z}_2$.
Let $f(x) = x^2 + x + 1$. This polynomial is irreducible over $\mathbb{Z}_2[x]$.
Therefore, all elements in $D=\mathbb{Z}_2[x]/(x^2+x+1)$ are invertible.
Consequently, $D$ is an integral domain with $|D| = 4$.