Show that a finite integral domain is a fieldSolutionThe mai

Solution

The main idea is that if R is finite, then any map from R to R is injective iff it is surjective iff it is bijective.

We know aR is a not a zero-divisor if fafa is injective. Since R is finite, then it is also surjective.

If fa is surjective then aa is a unit. Hence we have shown that any non-zero element in R is a unit. Hence R is a field.