Kähler differentials on a plane curve and a counterexample to a result of Zariski in positive characteristic

The aim of this article is to develop the theory of Kähler differentials on algebroid irreducible plane curves defined over algebraically closed fields of positive characteristic and to compare it with the theory over C. We emphasize here the study of the properties of the set of values of Kähler differentials, since they were an important invariant for the analytic classification of branches (cf. [8]), hoping that they will play an important role in positive characteristic too. At the end, we give a counterexample in positive characteristic for the result of Zariski that asserts that if a complex algebroid irreducible plane curve has coincident Tjurina and Milnor numbers, then the curve is equivalent to a monomial curve.