A Tschirnhausen transformation can be used to algebraically transform a general quintic equation to the form z^5 + c_1 z + c_0 = 0. In practice, the general quintic is first reduced to the principal quintic form y^5 + b_2 y^2 + b_1 y + b_0 = 0 before the transformation is done. Then, we require that the sum of the third powers of the roots vanishes, so s_3(y_j) = 0. We assume that the roots z_i of the Bring-Jerrard quintic are related to the roots y_i of the principal quintic form by z_i = α y_i^4 + β y_i^3 + γ y_i^2 + δ y_i + ϵ.

Bring quintic form | principal quintic form | quintic equation