Square-free Algorithms in Positive Characteristic

We study the problem of the computation of the square-free decomposition for polynomials over fields of positive characteristic. For fields which are explicitly finitely generated over perfect fields, we show how the classical algorithm for characteristic zero can be generalized using multiple derivations. For more general fields of positive characteristic one must make an additional constructive hypothesis in order for the problem to be decidable. We show that Seidenberg's Condition P gives a necessary and sufficient condition on the field K for computing a complete square free decomposition of polynomials with coefficients in any finite algebraic extension of K.