# a finite integral domain is a field

A finite integral domain is a field.

Proof:
Let $R$ be a finite integral domain. Let $a$ be nonzero element of $R$.

Define a function $\varphi\colon R\rightarrow R$ by $\varphi(r)=ar$.

Suppose $\varphi(r)=\varphi(s)$ for some $r,s\in R$. Then $ar=as$, which implies $a(r-s)=0$. Since $a\neq 0$ and $R$ is a cancellation ring, we have $r-s=0$. So $r=s$, and hence $\varphi$ is injective.

Since $R$ is finite and $\varphi$ is injective, by the pigeonhole principle we see that $\varphi$ is also surjective. Thus there exists some $b\in R$ such that $\varphi(b)=ab=1_{R}$, and thus $a$ is a unit.

Thus $R$ is a finite division ring. Since it is commutative, it is also a field.

Note:
A more general result is that an Artinian integral domain is a field.

Title a finite integral domain is a field AFiniteIntegralDomainIsAField 2013-03-22 12:50:02 2013-03-22 12:50:02 yark (2760) yark (2760) 11 yark (2760) Theorem msc 13G05 FiniteRingHasNoProperOverrings