Abstract: We revisit the classical problem of simplifying polynomials by means of Tschirnhaus transformations. We consider Tschirnhaus transformations involving (i) no auxiliary radicals, (ii) arbitrary radicals, (iii) odd degree radicals, and (iv) odd degree radicals and the square root of the discriminant. We previously showed that by using substitutions of type (i) one cannot reduce the general polynomial of degree n to a form with less than [n/2] independent coefficients. In this paper we give a new proof of this result and also extend it to transformations of types (iii) and (iv). In the last section we present alternative proofs, based on the cohomological approach shown to us by J.-P. Serre.