## 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, $n|m$.