Algebra & Number Theory
- Algebra Number Theory
- Volume 12, Number 8 (2018), 1823-1886.
On the relative Galois module structure of rings of integers in tame extensions
Let be a number field with ring of integers and let be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group of that involves applying the work of McCulloh in the context of relative algebraic theory. For a large class of soluble groups , including all groups of odd order, we show (subject to certain mild conditions) that the set of realisable classes is a subgroup of . 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.
Algebra Number Theory, Volume 12, Number 8 (2018), 1823-1886.
Received: 28 August 2015
Revised: 23 March 2018
Accepted: 2 July 2018
First available in Project Euclid: 21 December 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10]
Secondary: 11R32: Galois theory 11R65: Class groups and Picard groups of orders 19F99: None of the above, but in this section
Agboola, Adebisi; McCulloh, Leon R. On the relative Galois module structure of rings of integers in tame extensions. Algebra Number Theory 12 (2018), no. 8, 1823--1886. doi:10.2140/ant.2018.12.1823. https://projecteuclid.org/euclid.ant/1545361461