The gap between the Schur group and the subgroup generated by cyclic cyclotomic algebras

TL;DR

This paper investigates the relationship between the Schur group of an abelian extension of the rationals and the subgroup generated by cyclic cyclotomic algebras, providing a characterization of when this subgroup has finite index.

Contribution

It offers a characterization of when the subgroup generated by cyclic cyclotomic algebras has finite index in the Schur group based on the position of the extension in the cyclotomic lattice.

Findings

Finite index occurs under specific relative positions in the cyclotomic extension lattice.

Provides criteria for the subgroup's finite index in terms of cyclotomic extension structure.

Enhances understanding of the algebraic structure of Schur groups in number theory.

Abstract

Let $K$ be an abelian extension of the rationals. Let $S(K)$ be the Schur group of $K$ and let $CC(K)$ be the subgroup of $S(K)$ generated by classes containing cyclic cyclotomic algebras. We characterize when $CC(K)$ has finite index in $S(K)$ in terms of the relative position of $K$ in the lattice of cyclotomic extensions of the rationals.