“Principal Quintic Form”的概念、定义、习题解题方法及翻译-数学辞典

　

单词

Principal Quintic Form

释义

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 forthe s expresses them in terms of the Power sums . These Power sums can be expressed in termsof the s, so the s can be expressed in terms of the s. For a quintic to have no quartic or cubic term, thesums 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 usingNewton's Relations for the Power sums in the

, and finally solved. Therefore,

and

can beexpressed using Radicals in terms of the Coefficients

. Again by substitutioninto (4), we can calculate

,

and

in terms of

and

and the

. By theprevious 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

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更新时间：2025/2/22 2:15:39