Quintic equation

In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. For example:

Bring-Jerrard normal form

If

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:x^5+a_1x^4+a_2x^3+a_3x^2+a_4x+a_5=0,

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then if

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:y = x^4+b_1x^3+b_2x^2+b_3x+b_4,

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we may obtain a polynomial of degree five in y, a Tschirnhaus transformation, by for instance using the resultant to eliminate x. We might then seek particular values of the coefficients bi which make the coefficients for the polynomial for y of the form

Related Topics:
Tschirnhaus transformation - Resultant

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:y^5 + px + q,

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This reduction, discovered by Bring and rediscovered by Jerrard, is called Bring-Jerrard normal form. A direct attack on the reduction to Bring-Jerrard normal form does not work; the trick is to do it in stages, using more than one Tschirnhaus transformation, in which case modern computer algebra systems make the computations relatively easy.

Related Topics:
Bring - Jerrard

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We first note that substituting x5 - a1/5 in place of x removes the trace (degree four) term. We then may employ an idea due to Tschirnhaus to eliminate the x3 term also, by setting y = x2 + px + q and solving for p and q so as to eliminate the x4 and x3 terms both, we find that setting q = 2c/5 and

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: p = {sqrt{5c(3c^2-10d)} over 5c},

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eliminates both the third and fourth degree terms from

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:x^5 + cx^3 + dx^2 + ex + f,

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We now may successfully set

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:y = x^4+b_1x^3+b_2x^2+b_3x+b_4,

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in

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:x^5 + dx^2+ex+f,

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and eliminate the degree two term also, in a way which does not require the solution of any equation above degree three. This requires taking square roots for the values of b1, b2 and b4, and finding the root of a cubic for b3. The general form is easy enough to compute using a computer algebra package such as Maple or Mathematica, but is messy enough that it seems advisable to simply explain the method, which can then be applied in any particular case. However, it should be noted that what is entailed is a solution to the general quintic. In any particular case, one may set up the system of three equations, and then solve for the coefficients bi. One of the solutions so obtained will be as described, involving the roots of no polynomial higher than the third degree; taking the resultant with the coefficients so computed reduces the equation to Bring-Jerrard normal form. The roots of the original equation are now expressible in terms of the roots of the transformed equation.

Related Topics:
Square root - Cubic - Maple - Mathematica

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Regarded as an algebraic function, the solutions to

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:x^5+ux+v = 0,

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involves two variables, u and v, however the reduction is actually to an algebraic function of one variable, very much analogous to a solution in radicals, since we may further reduce the Bring-Jerrard form. If we for instance set

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:z = {x over (-u/5)^{1/5}},

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then we reduce the equation to the form

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:x^5 - 5x - 4t = 0,

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which involves x' as an algebraic function of a single variable t.

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