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).

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