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" $\displaystyle a$ and $\displaystyle b$ are not necessarily real numbers: they're elements of an arbitrary integral domain $\displaystyle R$.
So, pick $\displaystyle a\not =0$, and consider the function
$\displaystyle \phi :R\to R$
given by $\displaystyle \phi (r)=a\cdot r$. 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.