Finite Integral Domain is Galois Field

Theorem
A finite integral domain is a field.

Proof
To see this, observe that if $$a \ne 0$$, the mapping $$x \to a \circ x, x \in R$$ is injective because $$R$$ is an integral domain.

If $$R$$ is finite, the map is surjective as well, so that $$\exists x \in R: a \circ x = 1$$.