Given an absolutely irreducible representation of some finite group G, defined over some number field, the problem of finding fields affording the representation that are "small" is one of the core problems in the practical side of the representation theory. In this talk I will explain how Galois cohomology in the disguise of classification of central simple algebras can be used to decide if a field affords a given representation and, furthermore, to realise the representation over the new field. The main tools are the Hasse-Brauer-Noether theorem and explicit computation of finite group cohomology.