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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
::<math>z^5 + a\mu^4z + b\mu^5 = 0\,</math>
:with rational coefficients must satisfy the simple quadratic curve
::<math>y^2 = (20-a)(5+a)\,</math>
:for some rational <math>a, y</math>.

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]. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 09:59, 20 May 2015 (UTC)

:There is another option:

:::<math>{x}^{5}+k_{2}\,x+k_{1}=0</math>

:::<math>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}}</math>

::<math>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}}</math>

:::<math>m=\frac{{b}^{3}\,c\,g-b\,{c}^{4}}{{b}^{2}\,{g}^{3}+{c}^{3}\,{g}^{2}}</math>

:solution:

:::<math>g\,{m}^{\frac{4}{5}}-\frac{b\,g\,{m}^{\frac{3}{5}}}{c}+c\,{m}^{\frac{2}{5}}+b\,{m}^{\frac{1}{5}}</math>

:Sorry, I'm probably writing in the wrong section, but my additions were removed (I'm new). <small><span class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:A.Samokrutov|A.Samokrutov]] ([[User talk:A.Samokrutov|talk]] • [[Special:Contributions/A.Samokrutov|contribs]]) 16:35, 21 May 2015‎</span></small><!-- Template:Unsigned -->

::{{to|A.Samokrutov}} Please sign your posts in the talk pages with four tildes (<nowiki>~~~~</nowiki>).
::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. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 20:56, 21 May 2015 (UTC)

:Please look [https://oeis.org/A247344/a247344_1.txt quintic], [https://oeis.org/A255983/a255983_1.txt sextic], [https://oeis.org/A255985/a255985_1.txt septic] maybe something fit for a wiki? [[User:A.Samokrutov|A.Samokrutov]] ([[User talk:A.Samokrutov|talk]]) 16:30, 22 May 2015 (UTC)

::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  ''k''<sub>1</sub> and ''k''<sub>2</sub> or of the rational root of the Cayley's resolvent in terms of ''b'', ''c'', ''g''. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 17:19, 22 May 2015 (UTC)

==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?). --[[Special:Contributions/104.132.34.86|104.132.34.86]] ([[User talk:104.132.34.86|talk]]) 23:36, 15 February 2016 (UTC)

:{{fixed}}. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 09:32, 16 February 2016 (UTC)

== 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? [[User:Loraof|Loraof]] ([[User talk:Loraof|talk]]) 21:13, 13 June 2016 (UTC)
::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. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 22:29, 13 June 2016 (UTC)

== External links modified ==
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I have just modified {{plural:1|one external link|1 external links}} on [[Quintic function]]. Please take a moment to review [https://en.wikipedia.org/w/index.php?diff=prev&oldid=730883237 my edit]. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit [[User:Cyberpower678/FaQs#InternetArchiveBot|this simple FaQ]] for additional information. I made the following changes:
*Added archive https://web.archive.org/web/20140701060849/http://library.wolfram.com/examples/quintic/ to http://library.wolfram.com/examples/quintic/

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== Interwiki links recently disappeared ==
This page in english is now an [https://www.wikidata.org/wiki/Q33104507 orphan on wikidata], no more linked to the [https://www.wikidata.org/wiki/Q768390 corresponding pages in other languages]. [[User:Anne Bauval|Anne Bauval]] ([[User talk:Anne Bauval|talk]]) 22:21, 9 August 2017 (UTC)
: {{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. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 08:31, 10 August 2017 (UTC)
::You fixed the problem here (made the corresponding pages in other languages visible from here), and so did I [[fr:Special:Diff/139630566|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, [[de:Gleichung fünften Grades|the page in german]]. [[User:Anne Bauval|Anne Bauval]] ([[User talk:Anne Bauval|talk]]) 08:04, 20 August 2017 (UTC)

==Function vs. polynomial==
The name "quintic polynomial" is [https://books.google.com/ngrams/graph?content=quintic+polynomial%2Cquintic+function&year_start=1900&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cquintic%20polynomial%3B%2Cc0%3B.t1%3B%2Cquintic%20function%3B%2Cc0 much more common in Google ngrams] than "quintic function". It's also more explicit. I'll move the article unless I hear objections. --[[User:Macrakis|Macrakis]] ([[User talk:Macrakis|talk]]) 22:16, 29 October 2017 (UTC)
: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. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 07:37, 30 October 2017 (UTC)

== 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, 
[[User:Quinctic reader|Quinctic reader]] ([[User talk:Quinctic reader|talk]]) 19:44, 22 December 2017 (UTC) K. S. UYANIK (53, Turkish citizen in Ankara city of TURKEY.)
:{{to|Quinctic reader}} The article says: {{tq|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. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 20:46, 22 December 2017 (UTC)

== 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.
[[Special:Contributions/173.243.178.8|173.243.178.8]] ([[User talk:173.243.178.8|talk]]) 22:58, 14 June 2018 (UTC)

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

== Section on Hermite's work ==

{{u|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. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:22, 13 August 2023 (UTC)

:Wait a second, are you the one who wrote this? https://web.archive.org/web/20060117230549/http://www-calfor.lip6.fr/~dl/quinticV.5 ? Did you solve the issue where it says "this algorithm is not proved on a field containing sqrt(-1)" [[User:Valery Zapolodov|Valery Zapolodov]] ([[User talk:Valery Zapolodov|talk]]) 11:54, 8 March 2025 (UTC)
::This software is the test implementation of "Lazard, D. (2004). Solving quintics by radicals. ''The Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo, 2002'', 207-225". AFAIK, this is the first algorithm for explicitely solving algebraically ''all'' solvable quintics over a field of characteristic different of 2, 3, 5.
::I do not know which issue your are talking of, who is "it" in "it says", and of which algorithm you are talking of. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 12:33, 8 March 2025 (UTC)
:::It is a comment in your Maple code, it still works 22 years later. We also reimplemented it in Mathematica 14.2 where it works even better, the radicals of the roots are simplified totally, and not partially like in Maple. Have you ever implemented your 2014 paper, by the way? [[User:Valery Zapolodov|Valery Zapolodov]] ([[User talk:Valery Zapolodov|talk]]) 02:34, 15 March 2025 (UTC)

== 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[https://zh.wikipedia.org/zh/%E4%BA%94%E6%AC%A1%E6%96%B9%E7%A8%8B#]? You could add the resulting coefficient after each tranformation but it's very long [[User:ThinhDinh123|ThinhDinh123]] ([[User talk:ThinhDinh123|talk]]) 14:19, 24 October 2024 (UTC)

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