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

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- by Henry W. Thwing and Nickolas Heerema
- Trans. Amer. Math. Soc.
**195**(1974), 277-290 - DOI: https://doi.org/10.1090/S0002-9947-1974-0387259-3
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## 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.

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## Bibliographic Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**195**(1974), 277-290 - MSC: Primary 12F15
- DOI: https://doi.org/10.1090/S0002-9947-1974-0387259-3
- MathSciNet review: 0387259