A general quintic equation
(1)
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can be reduced to one of the form
(2)
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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)
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(4)
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Assume that the roots of the new quintic are related to the roots of the original quintic by
(5)
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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 .