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 $$f: x \mapsto a x$$.

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.