While reading a paper about the Kronecker-Weber Theorem, I noticed a theorem saying that for a Galois extension $K/\mathbb{Q}$, its Galois group is generated by $I_p$s, being the inertia groups of primes $p$ that ramify in $K$.
In the same paper however, they define the inertia group $I_P$ as depending on the prime $P$ that lies over $p$, so choosing a different $P$ gives another inertia group.
How should I interpret this ? That it doesn't matter which $P$ you choose, each $I_P$ will do ?
Any help would be appreciated.