  We consider a five-dimensional Einstein--Cartan spacetime upon which Dirac
spinor fields can be defined. Dirac spinor fields in five and four dimensions
share many features, like the fact that both are described by four-component
spinor fields, but they are also characterized by strong differences, like the
fact that in five dimensions we do not have the possibility to project on
left-handed and right-handed chiral parts unlike what happens in the
four-dimensional instance: we conduct a polar decomposition of the spinorial
fields, so to highlight all similarities and discrepancies. As an application
of spinor fields in five dimensions, we study Bianchi-I spacetimes, verifying
whether the Dirac fields in five dimensions can give rise to inflation or
dark-energy dominated cosmological eras or not.
