I am having troubles with the following problem:
Show that there is an integral domain with exactly 4 elements.
Any suggestion?
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 $\displaystyle |D| = 4$.