Abstract
Abstract In this article, we investigate the modified F ( T ) gravity, which is non-minimally coupled with the Dirac (fermion) field in Friedmann-Robertson-Walker space-time. Point-like Lagrangian is derived and modified Friedmann equations and Dirac equations for the fermion field are obtained by using the Lagrange multiplier. The Noether symmetry method related to differential equations is a useful tool for studying conserved quantities. In addition, this method is very useful for determining the unknown functions that exist in the point-like Lagrangian. Using this method, the form of the coupling between gravity and matter, the self-consistent potential, the symmetry generators, the form of F ( T ) gravity and the first integral (Noether charge) or a conserved quantity for this model are determined. Cosmological solutions that have a power-law form and describe the late time accelerated expansion of the Universe are obtained.