> [!NOTE] Lemma > Let $F$ be a [[Field (Algebra)|field]]. Then the [[Ring of Polynomial Forms|ring of polynomial forms]] over $F$, $F[X]$, is a [[Euclidean Domain|Euclidean domain]]. ###### Proof By [[Ring of Polynomial Forms over Integral Domain is Integral Domain]], $F[X]$ is indeed an integral domain. (EF2) follows from [[Division with Remainder Theorem for Ring of Polynomial Forms over Fields]]. (EF1) follows from [[Degree of Product of Polynomials Over Integral Domain]].