  The construction of fractional derivatives with the right properties for use
in field theory is reputed to be a difficult task, essentially because of the
absence of a unique definition and uniform properties. The conformable
fractional derivative introduced in 2014 by Khalil et al. in their seminal
paper is a novel and well-behaved definition of fractional derivative for a
function which is derivable in the usual sense. In this paper, we investigate
the consistency of the Euler-Lagrange formalism for a field theory defined on
such a fractional space-time. We especially focus on the relation between
symmetries and conservation laws (Noether's currents), about the symmetry group
introduced to construct the Lagrangian of the field. In particular, we show
that the use of the conformable derivative induces additional terms in the
calculation of the action variation. We also investigate the conservation of
the Noether current and show that this property only takes place on condition
that the equations of motion are verified with a new definition of the
conserved law.
