Use Gauss’ lemma to prove the Rational Root Theorem: If is a Euclidean domain with fraction field and if is a polynomial in , then any root of in has the form where and .
Suppose is a root of in in lowest terms (i.e. and are relatively prime). Now , so that (multiplying both sides by ) .
Now . Since divides each term on the right hand side, for some . Since and are relatively prime, and are relatively prime. Thus , by a previous exercise.
Similarly, for some , and we have .