Short proofs of some results of algebraic independence in non-zero characteristic.

The aim of this note is to give short and almost elementary proofs of two theorems, by Papanikolas and Chang-Yu, on algebraic independence of Carlitz logarithms and values of Carlitz-Goss zeta function, modifying and generalising arguments of Denis which proved earlier special cases of these results. These proofs where sketched in the text [12] and this note is intended to accompany it, somewhat as an informal appendix, by giving full details to some few lines remarks