## Characterize the ideals in F[x] principally generated by monic irreducibles

Let $F$ be a field, let $E$ be an extension of $F$, and let $\alpha \in E$ be algebraic over $F$ with minimal polynomial $p(x)$. Prove that the ideal $(p(x))$ in $F[x]$ is equal to $\{ q(x) \in F[x] \ |\ q(\alpha) = 0 \}$.

If $t(x) \in (p(x))$, then $t(x) = p(x)s(x)$ for some $s(x) \in F[x]$. Thus $t(\alpha) = p(\alpha)s(\alpha) = 0$.

Conversely, if $t(\alpha) = 0$, then the minimal polynomial $p(x)$ divides $t(x)$. So $t(x) \in (p(x))$.