$K/F$ is a Galois extension of algebraic number field, $\mathfrak{p}_{i}$ are prime ideals of $K$, and $\mathfrak{p}= \mathfrak{p}_{1}$.
Decompose a prime ideal $\mathcal{p}$ of $F$ in $K$. $$p(=po_K)=\prod_{i=1}^{g}(\mathfrak{p}_{i})^e$$ The decomposition group of $ \mathfrak{p}:Z=${$\sigma\in Gal(K/F)|\sigma\mathfrak{p}=\mathfrak{p}$}
The decomposition field of $\mathfrak{p}:K_{Z}=${$\alpha\in K|\sigma\alpha=\alpha, \forall\sigma\in Z$}
Firstly we have
1.#$Gal(K/F)=g$#$Z$ by $Gal(K/F)=\cup_{i=1}^{g}Z\sigma_{i}$, $\sigma_{i}$ depend on $\sigma_{i}\mathfrak{p}$ are all distinct prime ideals.
2.$[K:F]=efg$, $e$ is the ramification index, $f=[o_{K}/\mathfrak{p}_{i}:o_{F}/\mathcal{p}]$ is the inertia degree(independent of $i$ ).
then my professor’s note said we have $[K:K_{Z}]$=#$Z$，i want to know how to get the equation