Realization of all abelian number fields as character-value fields

Determine whether every abelian number field over the rationals is equal to Q(G) for some finite group G, where Q(G) is the field generated by all ordinary irreducible character values of G.

Background

For a finite group G, the paper defines Q(G) as the field generated over the rationals by the values of all irreducible ordinary characters of G on all elements of G. The paper studies this field for several families of finite linear, unitary, and simple groups, and uses those computations to classify fields of degree two or three arising from non-abelian simple groups.

The authors identify a broader unresolved realization problem: whether every abelian number field can arise as Q(G) for some finite group. This question is presented as remaining open and is attributed to an earlier question in the closing remarks of Feit and Guralnick, as well as to Navarro’s survey of open problems.