Let $k$ be a number field and $K / \mathbb Q$ a Galois extension containing $k$, with Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ and let $G_k:=\operatorname{Gal}(K/k)$. Let $\chi$ denote the character of the permutation representation $\rho$ of $G$ in $G/G_k$. How do I show the identity: $$\det[I-p^{-s}\rho(\sigma_p)|V_{p, \rho}] = \prod_{\mathfrak{p}|p} (1-N(\mathfrak{p})^{-s})$$ for every integer prime $p$? Here the determinant on the left is the Euler factor at prime $p$ appearing in the definition of the Artin $L$-function of the character $\chi$ and the product on the right is over all primes $\mathfrak{p}$ of $k$ which lie over $p$.
Edit: So (unaware of crossposting policies) I have already asked this question, - or rather a result that follows from the above one, - on Math Overflow (https://mathoverflow.net/questions/360714/on-l-function-of-permutation-representation), but I didn't really understand a lot of the details of the arguments in the answer I received there. Furthermore, I was hoping for a more elementary proof (if possible, something that probably goes along the lines of expanding the determinant as a product of eigenvectors). Also, I would really appreciate it if I could get the details of both the ramifying primes and unramifying primes cases (since I am not very comfortable with Artin $L$-functions). Thank you.