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)).

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