This is one of the classics.
Lemma. Let be a finite commutative ring. Then every element is either a unit or a zero-divisor.
Proof. If is not a zero-divisor, then the map is injective. Since is finite, it is also surjective, so there exists with .
Corollary 1. Let be a finite commutative ring. Then is its own total ring of fractions.
Proof. The total ring of fractions is the ring , where is the set of non-zerodivisors. But that set consists of units by the lemma above, so inverting them doesn’t change .
Corollary 2. Let be a finite domain. Then is a field.
Proof. In this case, the total ring of fractions is the fraction field. Therefore, is its own fraction field by Corollary 1.