I am reviewing some past exam questions and I have a problem solving the following example: Let $F = \mathbb{Q}(\zeta_5, \sqrt[5]{75})$, a field of degree 20 over $\mathbb{Q}$. Determine the decomposition and inertia groups in $Gal(F/\mathbb{Q})$ for primes in F above $3, 5, 11, 89$. Find the number of primes above $3, 5, 11, 89$ in the subfield $\mathbb{Q}(\sqrt[5]{75})$, tgether with their ramification and residue degrees over $\mathbb{Q}$.
My attempt: We first check the decomposition of the primes in $\mathbb{Q}(\sqrt{\zeta_5})$, using the Kummer-Dedekind criterion and can completely determine their factorization. My problem lies in checking the factorization in the field above. We could possibly check the how the polynomial $x^5-75$ decomposes modulo those primes and use again the same criterion(even though we would have to know things about the ring of integers of $\mathbb{Q}(\sqrt[5]{75})$).
Any help would be greatly appreciated!