Problem Statement: Show that a finite integral domain is a field.
Proof: Fix and consider the function for all . We wish to show that is injective, and thus has an inverse.
Suppose that , then which implies that . Since we are in an integral domain we know there are no zero divisors, so either or . We know that so it must be that which implies that and so is injective. Furthermore, since maps to , and is finite, we know that is onto . So, there is some such that , thus, .
Thus, is a field.
Reflection: This is a classic qual problem. It’s very straightforward and there are many ways to approach the proof. The key rests in the fact that is finite, otherwise we wouldn’t be able to say that is onto even though it maps to and is one-to-one.