Realizing Root Systems, their Weyl Groups, and Root Lattices in Number Fields

As is known, the set of all algebraic integers with norm 1 and 3 in a cyclotomic field generated by a cubic root of unity, is the root system of type $G_2$. The talk is aimed to discuss the possibility of realizing root systems, their Weyl groups, and root lattices in the form of objects naturally associated with number fields.