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The Quarterly Journal of Mathematics Advance Access originally published online on October 31, 2007
The Quarterly Journal of Mathematics 2008 59(2):149-162; doi:10.1093/qmath/ham037
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© 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

ON THE INTEGRAL GALOIS MODULE STRUCTURE OF CYCLIC EXTENSIONS OF p-ADIC FIELDS

Nigel P. Byott{dagger}

School of Engineering, Computing and Mathematics, University of Exeter, Exeter EX4 1QF

{dagger} E-mail: n.p.byott{at}ex.ac.uk

Received 25 August 2006; revised 30 April 2007
  Abstract

We strengthen results of Miyata on the integral Galois module structure of totally ramified cyclic Kummer extensions K of degree pn of a p-adic field k. Let c1(K/k) be the first ramification number of K/k, and let c(K/k) be the least non-negative residue of c1(K/k) modulo pn. Suppose that K is of the form k({alpha}) with {alpha}pn isin k and val K({alpha}–1)>0, (val K({alpha}–1), p)= 1. Then the valuation ring of K is free over its associated order A if c(K/k) divides pm–1 for some m with 1≤m≤n; the converse holds if n= 2; and A is a Hopf order (or a Gorenstein order) if and only if c(K/k) = pn–1.


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