  This paper investigates the fields of definition up to isogeny of the abelian
varieties called building blocks. A result of Ribet characterizes the fields of
definition of these varieties together with their endomorphisms, in terms of a
Galois cohomology class canonically attached to them. However, when the
building blocks have quaternionic multiplication, then the field of definition
of the varieties can be strictly smaller than the field of definition of their
endomorphisms. What we do is to give a characterization of the fields of
definition of the varieties in this case (also in terms of their associated
Galois cohomology class), by translating the problem into the language of group
extensions with non-abelian kernel. We also make the computations that are
needed in order to calculate in practice these fields from our
characterization.
