Show that every finite integral domain is a field.
Solution
Let \(R\) be a finite integral domain and let \(0\ne a\in R\). Consider multiplication by \(a\), \[ \mu_a:R\to R,\qquad x\mapsto ax. \] Because \(R\) has no zero divisors, \(\mu_a\) is injective. Since \(R\) is finite, it is therefore surjective. Hence there exists \(b\in R\) with \[ ab=1. \] Thus every nonzero element is invertible, so \(R\) is a field.