Fields of constants of integral derivations on a $p$-adic field

Abstract:

Let ${K_0}$ be a p-adic subfield of a p-adic field K with residue fields ${k_0} \subset k$. If ${K_0}$ is algebraically closed in K and k is finitely generated over ${k_0}$ then ${K_0}$ is the subfield of constants of an analytic derivation on K or equivalently, ${K_0}$ is the invariant subfield of an inertial automorphism of K. If (1) ${k_0}$ is separably algebraically closed in k, (2) $[k_0^{{p^{ - 1}}} \cap k:{k_0}] < \infty$, and (3) k is not algebraic over ${k_0}$ then there exists a p-adic subfield ${K_0}$ over ${k_0}$ which is algebraically closed in K. All subfields over ${k_0}$ are algebraically closed in K if and only if ${k_0}$ is algebraically closed in k. Every derivation on k trivial on ${k_0}$ lifts to a derivation on K trivial on ${K_0}$ if k is separable over ${k_0}$. If k is finitely generated over ${k_0}$ the separability condition is necessary. Applications are made to invariant fields of groups of inertial automorphisms on p-adic fields and of their ramification groups.