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.