Vol. 12, No. 8, 2018

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On the relative Galois module structure of rings of integers in tame extensions

Adebisi Agboola and Leon R. McCulloh

Vol. 12 (2018), No. 8, 1823–1886

Let F be a number field with ring of integers OF and let G be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group Cl(OFG) of OFG that involves applying the work of McCulloh in the context of relative algebraic K theory. For a large class of soluble groups G, including all groups of odd order, we show (subject to certain mild conditions) that the set of realisable classes is a subgroup of Cl(OFG). This may be viewed as being a partial analogue in the setting of Galois module theory of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups.

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Galois module structure, realisable classes, rings of integers, inverse Galois problem, relative K-group.
Mathematical Subject Classification 2010
Primary: 11R33
Secondary: 11R32, 11R65, 19F99
Received: 28 August 2015
Revised: 23 March 2018
Accepted: 2 July 2018
Published: 4 December 2018
Adebisi Agboola
Department of Mathematics
University of California, Santa Barbara
United States
Leon R. McCulloh
Department of Mathematics
University of Illinois
Urbana, IL
United States