## Principal Quintic Form

A general Quintic Equation

 (1)

can be reduced to one of the form
 (2)

called the principal quintic form.

Newton's Relations for the Roots in terms of the s is a linear system in the , and solving for the s expresses them in terms of the Power sums . These Power sums can be expressed in terms of the s, so the s can be expressed in terms of the s. For a quintic to have no quartic or cubic term, the sums of the Roots and the sums of the Squares of the Roots vanish, so

 (3) (4)

Assume that the Roots of the new quintic are related to the Roots of the original quintic by
 (5)

Substituting this into (1) then yields two equations for and which can be multiplied out, simplified by using Newton's Relations for the Power sums in the , and finally solved. Therefore, and can be expressed using Radicals in terms of the Coefficients . Again by substitution into (4), we can calculate , and in terms of and and the . By the previous solution for and and again by using Newton's Relations for the Power sums in the , we can ultimately express these Power sums in terms of the .