Galois splitting model of a local field (reviewed)

Let $K$ be a finite extension of $\Q_p$. A Galois splitting model of $K$ is an irreducible polynomial $f\in\Q[x]$ such that $K\cong \Q_p[x]/(f)$, and the Galois group of $f$ over $\Q$ is isomorphic to the Galois group of $f$ over $\Q_p$.

Most, but not all, local number fields have Galois splitting models.

In this case, the computation of various invariants related to $K$, such as the Galois invariants, can be computed more easily using $f$.