Minimizing representations over number fields II: Computations in the Brauer group

Finding minimal fields of definition for representations of finite groups is one of the most important unsolved problems of computational representation theory. While good methods exist for representations over finite fields, it is still an open question in the case of number fields. Continuing and extending previous work, we give a practical method for finding defining fields of minimal degree for absolutely irreducible representations. The method is based on techniques from Galois cohomology and the use of an explicit form of a weak Grunwald–Wang theorem.