Thread: Field

Let R be an integral domain having a finite number of elements. Prove R is a field.

Hint: Given any nonzero real number a, consider the list of elements ab for all real numbers b.

First of all, the "numbers" and are not necessarily real numbers: they're elements of an arbitrary integral domain .

So, pick , and consider the function



given by . Show that this function is injective (you will need to use the integral domain hypothesis), argue that it must thus also be surjective, and conclude.