## A monic irreducible polynomial over F is the minimal polynomial of its roots

Let $F$ be a field and let $p(x) \in F[x]$ be a monic irreducible polynomial. Let $E$ be an extension of $F$ containing the roots of $p(x)$. Suppose $\alpha \in E$ is a root of $p(x)$; prove that $p(x)$ is the minimal polynomial of $\alpha$ over $F$.

Since $\alpha$ is a root of $p(x)$, the minimal polynomial $M_{F,\alpha}(x)$ of $\alpha$ over $F$ divides $p(x)$ in $F[x]$. Since $p(x)$ is irreducible over $F$ and $M_{F,\alpha}$ is not a unit (since it has a root), $M_{F,\alpha}$ is a constant multiple of $p(x)$. Since both $p$ and $M_{F,\alpha}$ are monic, we have $p(x) = M_{F,\alpha}(x)$.