A general quintic equation
 |
(1)
|
can be reduced to one of the form
 |
(2)
|
called the principal quintic form.
Vieta's formulas 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 Vieta's
formulas 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 (◇), we can calculate
,
and 
in terms of 
and 
and the 
.
By the previous solution for 
and 
and again by using Vieta's
formulas for the power sums in the 
, we can ultimately express these power
sums in terms of the 
.
