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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Henry W. Thwing and Nickolas Heerema
Journal: Trans. Amer. Math. Soc. 195 (1974), 277-290
MSC: Primary 12F15
MathSciNet review: 0387259
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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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Keywords: Analytic derivation, integral derivation, field of constants, p-adic field, relative algebraic closure
Article copyright: © Copyright 1974 American Mathematical Society

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