Inertia group searching (awaiting review)

Let

• $K$ be a finite Galois extension of $\Q_p$ for some prime $p$,
• $\mathcal{O}_K$ the ring of integers for $K$,
• $P$ the unique maximal ideal of $\mathcal{O}_K$, and
• $k=\mathcal{O}_K/P$.

Then each $\sigma\in \Gal(K/\Q_p)$ induces a element of $\Gal(k/\F_p)$. The kernel of the resulting homomorphism \[ \Gal(K/\Q_p) \to \Gal(k/\F_p)\] is the inertia group of $K$.

When searching by inertia subgroups, it can be specified as an abstract group by either a GAP id or a group name from transitive group label.

It can also be specified by as a transitive group label. In this case, it will only match $p$-adic fields where the defining polynomial is irreducible over the unramified extension of $\Q_p$ of degree unramified degree and the Galois group of the polynomial over that extension matches the specified transitive group.