 p-adic field

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.