Computation of the Galois groups occuring in M. Papanikolas's study of Carlitz logarithms

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.