# Math Help - Field

1. ## 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.

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

So, pick $a\not =0$, and consider the function

$\phi :R\to R$

given by $\phi (r)=a\cdot r$. 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.