On Steinitz classes, realizable Galois module classes and embedding problems for non-abelian extensions of degree a power of 2

Abstract

Let Γ be the non-abelian group of order 16 which is the central product of the dihedral group of order 8 and the cyclic group of order 4, or an extra-special group of order 32. In the present article, we study an embedding problem connected with Steinitz classes in the perspective of studying the structure of the set of Galois module classes realizable by the rings of integers of extensions with Galois group isomorphic to Γ.

Keywords:

• Embedding problem
• Fröhlich–Lagrange resolvent
• Galois module structure
• locally free classgroups
• maximal order
• realizable classes
• rings of integers
• Steinitz classes

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 11R33

Acknowledgments

The authors are very grateful to the referee for his many detailed comments, his efficient and rigorous work despite the hard time due to Coronavirus.