On the degrees of the divisors of a composite polynomial

Let f(x) be an irreducible polynomial of degree n over a field F, and let g(x) be any polynomial in F[x]. Prove that every irreducible factor of the composite (f \circ g)(x) has degree divisible by n.

Let \beta be a root of f \circ g, and let m be the degree of \beta over F. Now g(\beta) \in F(\beta), and moreover g(\beta) is a root of the irreducible polynomial f(x). So F(g(\beta)) has degree n over F. Graphically, we have the following diagram of fields.

A field diagram

By Theorem 14 in D&F, n|m.

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