Precise examples for the solvable quintic with the “most complexity” of roots

algebra-precalculusgalois-theoryirreducible-polynomialsquinticsradicals

We know that, the "greater" the degree of a polynomial equation, the greater the "complexity" of the roots in general.

For example, the overall complexity of the roots of a general cubic equation is obviously more complicated than the general quadratic. However, the overall complexity of the roots of a general quartic equation is "greater" than any of them. We understand this when we look at the general formulas.

Motivation:

I'm wondering how much "more" the complexity of the roots of a solvable quintic equation can be than the general quartic.

But, here I would like to observe this "event" on the precise example quintic, rather than just Galois theory.

For example, I would like to "compare" a solvable "most complex" quintic with the complexity of the roots of the equation $$2x^4+5x^3+7x^2+13x+3=0:$$

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Then, I found the this solvable quintic:

$$x^5-5x+12=0$$

and the exact real root equals to:

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However, I do not have information that this is included in the examples I am looking for.

In short, I am looking for a precise example for the solvable quintic whose roots are the most complicated.

Of course, the example this type of quintic itself can be interesting. But, I would like to see its roots expressed in radicals.

At least I know that the equation I'm looking for will be irreducible.

Best Answer

TL, DR: It boggles the mind.

  1. If the quintic equation $x^5+ax^4+bx^3+cx^2+dx+e=0$ is solvable by radicals at all, the roots will appear in the form

$x=-a/5+\sqrt[5]{\rho_1}+\sqrt[5]{\rho_2}+\sqrt[5]{\rho_3}+\sqrt[5]{\rho_4},$

where $\rho_1,\rho_2,\rho_3,\rho_4$ are roots of a quartic equation. Thus to begin with you have four quartic roots, with their complexity, each under a fifth-root radical sign.

  1. The quartic equation must be derived by auxiliary calculations, which likely will produce additional layers of radicals. In general, this process should be expected to introduce numbers that are absolutely much larger than the input coefficients $a,b,c,d,e$ when the latter are integers.

  2. And now the coup de grace: The radical formula requires as input not only the coefficients of the quintic equation, but also a root of some resolvent sextic (several versions of which are avaiable). This resolvent sextic (1) has to be derived by still more operations from the quintic coefficients, (2) will again typically involve absolutely much larger numbers than the original coefficients $a,b,c,d,e$ when these are integers, and (3) must be searched for a rational root to provide this input, which will involve trial and error and succeed only for a minority of quintic equations.

Wikipedia gives a brief description of what is involved in the solution process for a "reduced" quintic equation

$y^5+py^3+qy^2+ry+s=0.$

The method presented there uses the resolvent sextic by Cayley. First we have to construct the resolvent and find a rational root, as described above. Then, if that succeeds:

Cayley's result allows us to test if a quintic is solvable. If it is the case, finding its roots is a more difficult problem, which consists of expressing the roots in terms of radicals involving the coefficients of the quintic and the rational root of Cayley's resolvent.

In 1888, George Paxton Young described how to solve a solvable quintic equation, without providing an explicit formula;[1] in 2004, Daniel Lazard wrote out a three-page formula.[2]

So ... for the specific case where the quartic term is reduced to zero, and if you fortunately have a rational root of the Cayley resolvent in hand, then you are to put the quintic coefficients and the resolvent root into a formula that runs to three pages and will involve large numbers for even relatively simple-looking quintic equations.

You may want to consider numerical solution methods.

Cited references

  1. Klein, Felix (1888). Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree. Translated by Morrice, George Gavin. Trübner & Co. ISBN 0-486-49528-0.

  2. Lazard, Daniel (2004). "Solving quintics in radicals". In Olav Arnfinn Laudal; Ragni Piene (eds.). The Legacy of Niels Henrik Abel. Berlin. pp. 207–225. ISBN 3-540-43826-2. Archived from the original on January 6, 2005.

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