Bring-Jerrard Quintic Form

A Tschirnhausen transformation can be used to algebraically transform a general quintic equation to the form

(1)

In practice, the general quintic is first reduced to the principal quintic form

(2)

before the transformation is done. Then, we require that the sum of the third powers of the roots vanishes, so . We assume that the roots of the Bring-Jerrard quintic are related to the roots of the principal quintic form by

(3)

In a similar manner to the principal quintic form transformation, we can express the coefficients in terms of the .