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
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 .
See also Bring Quintic Form, Newton's Relations, Quintic Equation
© 19969 Eric W. Weisstein
19990526