A Universe of Sorts

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 $p$ .

§ Irreducible Polynomial over a Field Divides Any Polynomial with Common Root