Root systems in number fields

TL;DR

This paper classifies root systems in number fields based on the inclusion of their Weyl groups in a group generated by automorphisms and multiplications, linking algebraic structures with number field properties.

Contribution

It provides a classification of root systems in rings of integers of number fields where Weyl groups are contained in a specific automorphism-related group.

Findings

Classified root systems with Weyl groups in the group generated by automorphisms and multiplications.

Identified Weyl groups of roots systems of rank n as subgroups of this group for degree n number fields.

Abstract

We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $\mathcal L(K)$ generated by ${\rm Aut} (K)$ and multiplications by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are isomorphic to a subgroup of $\mathcal L(K)$ for a number field $K$ of degree $n$ over $\mathbb Q$.