Finite Integral Domain is Galois Field

Theorem
A finite integral domain is a field.

Proof
Let $$R$$ be a finite integral domain.

Fix $$a\in R$$, $$a\ne 0$$. We wish to show that a has an inverse in $$R$$. So consider the map $$f:R\to R$$ defined by $$f:x\mapsto ax$$.

We first show that the kernel of $$f$$ is trivial. Consider that $$\ker(f)=\{x\in R:f(x)=0\}=\{x\in R:ax=0\}$$. Since $$R$$ is an integral domain, it has no zero divisors and thus $$ax=0$$ means that $$a=0$$ or $$x=0$$. Since, by definition, $$a\ne 0$$, then it must be true that $$x=0.$$ Therefore, $$\ker(f)=\{0\}$$ and so $$f$$ is injective.

Next, the Pigeonhole Principle gives us that an injective map from a finite set onto itself is surjective. Since $$R$$ is finite, the map $$f$$ is surjective.

Finally, since $$f$$ is surjective and $$1\in R$$, $$\exists\ x \in R: f(x) = ax = 1$$. So this $$x$$ is the inverse of $$a$$ and we are done.