  In this note, we state a theorem of compution of the unipotent radical of the
Galois group of an object $U$ of a tannakian category defined over a field of
positive characteristic, extension of the unit object by a semi-simple one. We
then give a criteria of algebricity of solutions of the objects in terms of
linear dependences in groups of extension. We apply these results to the
tanakian framework of $t$-motives developped by M. Papanikolas; in particular,
we give an alternative proof of the algebraic independence of Carlitz
logarithms.
