Weierstrass points on Kummer extensions

For Kummer extensions given by y(m) = f(x), we discuss conditions for an integer to be a Weierstrass gap at a place P. In the case of fully ramified places, the conditions are necessary and sufficient. As a consequence, we extend independent results of several authors. Moreover, we show that if the Kummer extension is F-q2-maximal and f(x) is an element of F-q2 [x] has at least two roots with the same multiplicity lambda coprime to m, then m divides 2(q + 1). Under the extra condition that either m or the multiplicity of a third root of f(x) is odd, we conclude that m divides q + 1.