New York Journal of Mathematics
Volume 19 (2013) 647-655


Paul J. Truman

Integral Hopf-Galois structures for tame extensions

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Published: October 9, 2013
Keywords: Hopf-Galois structures, Hopf-Galois module theory, Hopf order, tame ramification
Subject: 11R33 (primary), 11S23 (secondary)

We study the Hopf-Galois module structure of algebraic integers in some Galois extensions of p-adic fields L/K which are at most tamely ramified, generalizing some of the results of the author's 2011 paper cited below. If G=Gal(L/K) and H=L[N]G is a Hopf algebra giving a Hopf-Galois structure on L/K, we give a criterion for the OK-order OL[N]G to be a Hopf order in H. When OL[N]G is Hopf, we show that it coincides with the associated order AH of OL in H and that OL is free over AH, and we give a criterion for a Hopf-Galois structure to exist at integral level. As an illustration of these results, we determine the commutative Hopf-Galois module structure of the algebraic integers in tame Galois extensions of degree qr, where q and r are distinct primes.

Author information

School of Computing and Mathematics, Keele University, UK