§ Irreducible polynomial over a field divides any polynomial with common root

• Let $p(x) \in K[x]$ be an irreducible polynomial over a field $K$. Let $p$ it share a common root $\alpha$ with another polynomial $q(x) \in K[x]$. Then we claim that $p(x)$ divides $q(x)$.
• Consider the GCD $g \equiv gcd(p, q)$. Since $p, q$ share a root $\alpha$, we have that $(x - \alpha)$ divides $g$. Thus $g$ is a non-constant polynomial.
• Further, we have $g | p$ since $g$ is GCD. But $p$ is irreducible, it cannot be written as product of smaller polynomials, and thus $g = p$.
• Now, we have $g | q$, but since $g = p$, we have $g | q$. This implies $p | q$ for any $q$ that shares a root with $p$.