## In a polynomial ring over a field, greatest common divisors exist and are unique up to units and Bezout’s identity holds

Let $F$ be a field and let $a(x),b(x) \in F[x]$. Then a greatest common divisor $d(x)$ of $a(x)$ and $b(x)$ exists and is unique up to multiplication by a unit. Moreover, there exist $s(x)$ and $t(x)$ such that $d = as+bt$.

This follows because $F[x]$ is a Euclidean domain.