I am having troubles with the following problem:

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

Any suggestion?

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- Dec 2nd 2008, 10:03 AMynn6871Integral Domain
I am having troubles with the following problem:

__Show that there is an integral domain with exactly 4 elements.__

Any suggestion? - Dec 2nd 2008, 10:54 AMThePerfectHacker
Let $\displaystyle F = \mathbb{Z}_2[x]$ - the polynomials with coefficients in $\displaystyle \mathbb{Z}_2$.

Let $\displaystyle f(x) = x^2 + x + 1$. This polynomial is irreducible over $\displaystyle \mathbb{Z}_2[x]$.

Therefore, all elements in $\displaystyle D=\mathbb{Z}_2[x]/(x^2+x+1)$ are invertible.

Consequently, $\displaystyle D$ is an integral domain with $\displaystyle |D| = 4$.