I am a beginner student of Algebraic Number Theory and I am starting to learn ramification theory (of global fields). My question asks for motivation for a definition I was given.
Let $K$ be an algebraic number field, $\mathcal{O}_{K}$ its ring of integers, $L/K$ a Galois extension and $\mathcal{O}_{L}$ the integral closure of $\mathcal{O}_{K}$ in $L$.
I know that the group $G=Gal(L/K)$ acts transitively on the set of prime ideals $\mathfrak{P}_{i}$ of $\mathcal{O}_{L}$ above a prime $\mathfrak{p}$ of $\mathcal{O}_{K}$ and it's just a natural thing to consider the decomposition group (of one of these ideals) $G^{Z}(\mathfrak{P})=\{\sigma\in G\:|\:\sigma(\mathfrak{P})=\mathfrak{P}\}$, which is the stabilizer of $\mathfrak{P}$ under this action.
Now, in the paper I am following, together with the decomposition group, it was defined the group \begin{equation} G^{T}(\mathfrak{P})=\{\sigma\in G\:|\:\sigma(\alpha)\equiv\alpha\mod \mathfrak{P}\:\:\forall\alpha\in\mathcal{O}_{L}\}, \end{equation}
and this one I want to understand better.
I was told that each element of $G^{Z}(\mathfrak{P})$ induces an automorphism in the quotient $\mathcal{O}_{L}/\mathfrak{P}$, which is pretty reasonable. This $G^{T}(\mathfrak{P})$ looks like the subgroup of elements of $G^{Z}(\mathfrak{P})$ that induce the identity in the quotient $\mathcal{O}_{L}/\mathfrak{P}$. From my spying on other books and papers, i recognize this group as the so called $\textbf{Inertia group}$.
My question is basically:
What does the Inertia group tells us? When we look at the index $(G:G^{Z}(\mathfrak{P}))$, it gives us a notion of "how many primes did $\mathfrak{p}$ split into in $\mathcal{O}_{L}$". What about the inertia group? What is its meaning? And it is something as natural as considering the stabilizer of a group action?