Talk:Quintic function

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Tagged assertion

Before my recent edits, the article contained the following assertion, which I quote verbatim for the record:

It is then a necessary (but not sufficient) condition that the irreducible solvable quintic

z^{5}+a\mu ^{4}z+b\mu ^{5}=0\,

with rational coefficients must satisfy the simple quadratic curve

y^{2}=(20-a)(5+a)\,

for some rational

a,y

.

As it is, it is a mathematical non-sense. I have tried to give it a meaning by replacing y by b, and rewording the sentence, but this results in a wrong assertion (at least if there is no typo in the formula for Cayley's resolvent). As no source is provided, which allows to correct this sentence, I'll remove it as [[WP:OR]. D.Lazard (talk) 09:59, 20 May 2015 (UTC)[reply]

There is another option:

{x}^{5}+k_{2}\,x+k_{1}=0

k_{2}={\frac {5\,{b}^{2}\,\left({b}^{2}\,g-{c}^{3}\right)\,\left({b}^{4}\,{g}^{2}-{b}^{2}\,{c}^{3}\,g-{c}^{6}\right)\,\left({b}^{4}\,{g}^{2}+4\,{b}^{2}\,{c}^{3}\,g-{c}^{6}\right)}{c\,{g}^{3}\,{\left({b}^{2}\,g+{c}^{3}\right)}^{3}}}

k_{1}={\frac {{b}^{2}\,\left({b}^{2}\,g-{c}^{3}\right)\,\left({b}^{4}\,{g}^{2}+{c}^{6}\right)\,\left({b}^{8}\,{g}^{4}-22\,{b}^{6}\,{c}^{3}\,{g}^{3}-6\,{b}^{4}\,{c}^{6}\,{g}^{2}+22\,{b}^{2}\,{c}^{9}\,g+{c}^{12}\right)}{{c}^{2}\,{g}^{4}\,{\left({b}^{2}\,g+{c}^{3}\right)}^{4}}}

m={\frac {{b}^{3}\,c\,g-b\,{c}^{4}}{{b}^{2}\,{g}^{3}+{c}^{3}\,{g}^{2}}}

solution:

g\,{m}^{\frac {4}{5}}-{\frac {b\,g\,{m}^{\frac {3}{5}}}{c}}+c\,{m}^{\frac {2}{5}}+b\,{m}^{\frac {1}{5}}

Sorry, I'm probably writing in the wrong section, but my additions were removed (I'm new). — Preceding unsigned comment added by A.Samokrutov (talk • contribs) 16:35, 21 May 2015‎

To editor A.Samokrutov: Please sign your posts in the talk pages with four tildes (~~~~).

This formula seems a new parameterization of solvable quintics in Bring–Jerrard form. The article gives already three different such parameterizations, which are all simpler than yours. As Wikipedia is not a media for publishing original research (see WP:OR), this new parameterization cannot be inserted in Wikipedia if it has not been published in a referred and notable academic journal. Ever if it has been reliably published, it is not necessary worth to mention it in Wikipedia: this would need that it would be cited in secondary sources (see WP:SECONDARY), or, at least that it is evident that would improve the article. In any case, because of your conflict of interest (see WP:COI), you are misplaced to judge if this would improve the article. D.Lazard (talk) 20:56, 21 May 2015 (UTC)[reply]

Please look quintic, sextic, septic maybe something fit for a wiki? A.Samokrutov (talk) 16:30, 22 May 2015 (UTC)[reply]

Again, without a referred article explaining the meaning of these formulas, how there were obtained, and why they are interesting, these formulas may not be inserted in Wikipedia.

PS to my preceding post: There is no evidence that above parameterized set of quintics is solvable. A proof of solvability could be an expression of b, c, g in terms of k₁ and k₂ or of the rational root of the Cayley's resolvent in terms of b, c, g. D.Lazard (talk) 17:19, 22 May 2015 (UTC)[reply]

Depressing the quintic

Article reads: "the Tschirnhaus transformation ...which depresses the quintic". I bet most readers have no idea what it means to "depress" a quintic. Some more explanation would be helpful (or maybe even an article on depressing polynomials?). --104.132.34.86 (talk) 23:36, 15 February 2016 (UTC)[reply]

Fixed. D.Lazard (talk) 09:32, 16 February 2016 (UTC)[reply]

Quintic or irreducible quintic?

The section Quintic function#Roots of a solvable quintic (paragraph 2) says

If the quintic is solvable, one of the solutions may be represented by an algebraic expression involving a fifth root and at most two square roots, generally nested.

Should this start out as "If an irreducible quintic is solvable..."? I ask because a quintic that can be factored into a cubic and a quadratic can be solved without fifth roots—can it also be solved using a fifth root? Loraof (talk) 21:13, 13 June 2016 (UTC)[reply]

Every equation may be solved using a fifth root. It suffice to replace any number appearing in the solution by the fifth power of its fifth root. This may seem a joke, but Galois theory allows showing that fifth roots (if any) may always be eliminated from any solution of an equation of lower degree. The beginning of the section already said that "solvable quintics" refer to irreducible quintics solving by radicals. I have clarified this by adding that only irreducible quintics are considered in the section. D.Lazard (talk) 22:29, 13 June 2016 (UTC)[reply]

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Interwiki links recently disappeared

This page in english is now an orphan on wikidata, no more linked to the corresponding pages in other languages. Anne Bauval (talk) 22:21, 9 August 2017 (UTC)[reply]

Fixed. The problem is that somebody has decided that "Quintic function" should be different from "Quintic equation". Some time ago, I have encountered this problem with Fibonacci number and Fibonacci sequence. I have copied from there the mysterious way for fixing the problem. D.Lazard (talk) 08:31, 10 August 2017 (UTC)[reply]

You fixed the problem here (made the corresponding pages in other languages visible from here), and so did I on the page in french (made your page in english visible from there), but this page in english is still no longer visible from, for instance, the page in german. Anne Bauval (talk) 08:04, 20 August 2017 (UTC)[reply]

Function vs. polynomial

The name "quintic polynomial" is much more common in Google ngrams than "quintic function". It's also more explicit. I'll move the article unless I hear objections. --Macrakis (talk) 22:16, 29 October 2017 (UTC)[reply]

I agree. However, as there are 3 lines in the history, I guess that you should pass through a move request.

By the way, a similar move would be useful for cubic, quartic, and other, as in all these articles the "function" aspect is minor, and most of the content is devoted to the equations, which are studied using only polynomial technics. D.Lazard (talk) 07:37, 30 October 2017 (UTC)[reply]

The root z in the Cayley's resolvent

Unfortunately I am unable to find the relation between the root z in the Cayley's resolvent and the actual root y of the depressed quintic equation - the one lacking the biquadratic term in y! I scanned several times the related paragraphs in the main wiki-article but couldn't see a given relation. Can you help me see if there is any of such relation given in the article? If not, please add it immediately. I wait for your reply in my e-mail account. Yours truly, Quinctic reader (talk) 19:44, 22 December 2017 (UTC) K. S. UYANIK (53, Turkish citizen in Ankara city of TURKEY.)[reply]

To editor Quinctic reader: The article says: 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. This means that it is difficult to find the relationship between the rational root of Cayley's resolvent and the roots of the quintic. In fact, I have given a solution to this problem (cited in the article), and the description of the formula is three pages length. D.Lazard (talk) 20:46, 22 December 2017 (UTC)[reply]

Broken link or bad external reference

The first external link to the Quintic Equation Solver appears to be broken, or the solver isn't working. Page comes up blank. 173.243.178.8 (talk) 22:58, 14 June 2018 (UTC)[reply]

I have removed it. Thank you for pointing this out.—Anita5192 (talk) 23:35, 14 June 2018 (UTC)[reply]

Section on Hermite's work

Reformbenediktiner Insists to add a section specifically on Hermite's work on the subject. This section is not convenient for this Wikipedia article. Firstly, it seems to be an wp:original synthesis of several Hermite's articles. In any case, no WP:secondary sources are provided. Moreover, this editor describes (in an edit summary) his edit as "I made everything very carefully. Read the Hermite essay in the Italian version on page 258! These is this special formula. And I only did regular calculation algorithms we all know. And the last step I wrote down in this article just needs the theorems of the lemniscate elliptic functions that are also known already". This is exactly what is called "original research" in Wikipedia. So, the Wikipedia policy WP:NOR implies to remove this new section.

There are other reasons to remove this section: firstly, notations are not defined, nothing is said on the relation of the first formula with the general quintic, and the final result is not explained. This is not acceptable for an encyclopedia. Also, Hermite's work is mentioned in the preceding section, which refers to Bring radical for details. In the linked article, the relation of the subject with elliptic integrals is detailed and the other authors who have contributed are mentioned.

In other words, this new section is a poor duplication of an existing content. This is a further reason for removing it. D.Lazard (talk) 15:22, 13 August 2023 (UTC)[reply]

About the Solving with Bring radicals section

Maybe add things on how to reduce the general quintic to the Bring-Jerrard normal form? Like in the Chinese version of this wiki[1]? You could add the resulting coefficient after each tranformation but it's very long ThinhDinh123 (talk) 14:19, 24 October 2024 (UTC)[reply]

The section § Solving with Bring radicals presented as a summary of the article Bring radical by the template {{Main article|Bring radical}} placed at the top of the section. So, the details that you are asking for are normally in that article. D.Lazard (talk) 14:36, 24 October 2024 (UTC)[reply]