Irreducible polynomial over normal extension

Question

Let $E/F$ be a normal extension and a irreducible polynomial $f(X)∈F[X]$.

Prove that all irreducible factors of $f(x)$ in $E[X]$ have the same degree.

More explicit: if $f$ factors as the product of irreudiclbe polynomials $g_1\cdots g_k$ over $E$, then $deg(g_i)=deg(g_j)$ for all $i\neq j$.

I've tried to use that hint: On irreducible polynomial over normal extension

But i don't understand how to use these facts.

Answer 1

If we let $f(x)\in F[X]$ be irreducible, it has a root in $E$ if and only if it has a linear factor in $E[X]$. If $f$ has a root, this means that all irreducible factors of $f$ in $E[X]$ must be linear, making $f$ split over the field $E$ (hence, the definition of being a normal field extension).