p-adic field ($[K:{\mathbb{Q}}_p]<\infty$)

A p-adic field is a finite extension of the P-adic number field ${\mathbb{Q}}_p$, equipped with its extended p-adic absolute value. Conversely every characteristic-zero, nontrivially valued, non-Archimedean locally compact field is such an extension. A finite quotient $\mathcal{O}/p\mathcal{O}$ supplies finitely many generators of $\mathcal{O}$ over ${\mathbb{Z}}_p$ by repeated reduction and completeness.

An integral lattice in a finite-dimensional ${\mathbb{Q}}_p$-vector space is a free ${\mathbb{Z}}_p$-submodule of full rank. Any two such lattices are commensurable: their intersection has finite index in each, because a sufficiently large power of $p$ multiplies either into the other. A finite group acting on the space admits invariant lattices by taking a sum of translates. This allows Herbrand quotient comparisons via finite quotient modules.