Fields of constants of integral derivations on a $p$-adic field
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- by Henry W. Thwing and Nickolas Heerema PDF
- Trans. Amer. Math. Soc. 195 (1974), 277-290 Request permission
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.References
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Additional 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