Integral Hopf-Galois structures for tame extensions

Truman, Paul

Abstract

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.

Citation

Truman, P. (2013). Integral Hopf-Galois structures for tame extensions

• Acceptance Date: Oct 9, 2013
• Publication Date: Oct 9, 2013
• Journal: New York Journal of Mathematics
• Print ISSN: 1076-9803
• Pages: 647-655
• Keywords: Hopf-Galois structures; Hopf-Galois module theory; Hopf order; tame ramification
• Publisher URL: http://nyjm.albany.edu/j/2013/19-32.html