Talk:Bring radical

message: The section really needs to be expanded. There has to be some sort of discussion of the original Hermite-Kronecker method which uses elliptic modular functions, as well as the method outlined by Felix Klein in his Lectures on the Icosahedron. These are the original methods for solving the quintic developed in the nineteenth century. --Stormwyrm (talk) 21:15, 19 September 2008 (UTC)
thread_title: Other Characterizations of the Bring Radical

message: Are there any results for higher polynomials? For example, is it possible to solve a degree 6 polynomial without introducing any further radicals? (This seems plausible, perhaps degree 6 polynomials are either a quintic in disguise, or a cubic of a quadratic) --njh 12:16, 12 July 2006 (UTC)
thread_title: Extensions to higher order polynomials

message: The link to "radicals" in the third line of this article should be made more specific. Presumably its using one of the four meanings below, but I have no idea which one:
* Radical of an algebraic group, a concept in algebraic group theory
* Radical of an ideal, an important concept in abstract algebra
* Radical of an integer, a concept in number theory
* Radical of a bilinear form, a concept in linear algebra
Could someone who knows fix the link to make it more useful to those who haven't taken math since high school like myself?
Thanks, mennonot 15:44, 8 November 2006 (UTC)
thread_title: Radicals Disambiguation needed

message: I couldn't find much agreement above. Perhaps a new discussion section can yield more of a consensus.
What I found to be the high points of discussion:
 	Delete                                                                                                                                                                                                                            	Keep
1	Original artwork. Wikipedia is not a publisher of original cartoons.                                                                                                                                                              	Almost all artwork on Wikipedia is original artwork, since Wikipedia requires free images.
2	Serious encyclopedias don't use cartoons to illustrate points.                                                                                                                                                                    	Many serious books (textbooks et al) use cartoons.
3	In popular culture sections are lists of factual data, whereas this joke is pure frivolity.                                                                                                                                       	Wikipedia articles have In popular culture section, precedent for including (relevant) frivolity.
4	Does not add or clarify any information in the article; unencyclopedic.                                                                                                                                                           	Pun reiterates that Bring radicals are needed to solve quintics.
5	Teachers use all kinds of pedagogic tools that do not belong in encyclopedias. Wikipedia is not a textbook / learning aid, but solely an information source. This image is a helpful in remembering, not in conveying information.	A "serious" work can use cartoons to illustrate a point. Something that is useful to the reader should be included. There is much on Wikipedia does not strictly convey factual information, but make learning more pleasurable and reinforce material.
I think issues 4 and 5 are most important. The only content explained by this cartoon is that "Bring radicals are needed to solve quintic equations." Why take a large image to reiterate, in a vague manner, a simple concept that can be expressed in a single sentence? ~ Booya Bazooka 11:50, 16 July 2007 (UTC)
replies: text: Great synopsis and good point. However I think that maybe this should be tabled until the article itself is improved. There have been some concerns raised at WT:WPM about the contents of the article (gasp) :) And I think that those concerns should take precedence.--Cronholm144 12:54, 16 July 2007 (UTC)
replies: text:  A valid concern, although I don't think this discussion is really hindering development to the rest of the article. Also, I'd really just like to know where policy stands on this, since the question will inevitably come up again elsewhere. ~ Booya Bazooka 14:01, 16 July 2007 (UTC)

text: This cartoon is a good example for debate since it serves a purpose in reinforcing and making more memorable the main idea. My main problem with it is that, as opposed to a graph or diagram, its information content is very low and it does not present information in a fundamentally different or visual way. The appearance of the speakers is completely arbitrary and it might as well be written as three lines of dialogue in text (which would eat up a lot less space). Nevertheless I wouldn't fight very strongly for its removal. Dcoetzee 22:04, 16 July 2007 (UTC)

text: Why not send this image over to Wikibooks, where they will actually use it? This is meant for a textbook, not an encyclopedia.-Wafulz 17:47, 16 August 2007 (UTC)
replies: text: I agree. Encyclopedias are not meant to be teachers or textbooks. We shouldn't employ shallow devices aimed only at capturing readers' attention. The info is in the caption; the joke picture adds no information. nadav (talk) 20:14, 17 August 2007 (UTC)

text: The is absolutely no prerequisite whatsoever for expressing an opinion on whether this cartoon should be included, except being a wikipedian in good standing. nadav (talk) 21:09, 17 August 2007 (UTC)
replies: text: Exactly Nadav. I have no experience with the math involved in this article, but the pun was readily apparent, don't think that it's just math majors who understand it. Nonetheless, the picture is an original work, does not illustrate the subject in any meaningful way, and provides no new information. --YbborTalk 21:37, 17 August 2007 (UTC)

text: The cartoon image was kept mainly because it was properly licensed (and this has been overturned at deletion review). The point remains that an encyclopedia should not employ cutesy cartoons on articles in an attempt to make them funnier. >Radiant< 09:27, 22 August 2007 (UTC)

text: Keep. It is funny. Besides, it does actually serve a pedagogical purpose in that it, on first reading, seems to contradict a known fact about solving quintics, but then after the pun is recognized, emphasizes distinction between types of solvability. --C S (Talk) 07:29, 18 March 2006 (UTC)

text: Delete, no fugly drawings on Wikipedia, please. I'd not take serious an encyclopedia article with that sort of illustration. Also not funny for those lacking an university education in math, i.e. 99% of the world. Sandstein 14:45, 18 March 2006 (UTC)

text: Keep. It is in use on someone's page, I see no harm in keeping it. Weatherman90 17:17, 18 March 2006 (UTC)
replies: text: Er, it says: "No pages link to here." Sandstein 18:29, 18 March 2006 (UTC)
replies: text: Check again. User:CyborgTosser
--Weatherman90 22:14, 18 March 2006 (UTC)
replies: text: Oh, now I see. I looked here. Still, I think it's just not done well enough to illustrate a scientific article. If you just want to keep it on your user page, no problem. Sandstein 07:17, 19 March 2006 (UTC)
replies: text: That was actually me who said "Check again", my signature got misplaced somehow. You must have read the userpage link that I pasted on there as a signature. The user who has it on his page has yet to comment. Weatherman90 16:04, 19 March 2006 (UTC)

text: Keep. To summarize my comments on Talk:Quintic equation, 1. silly cartoons are found in serious encyclopedias (although I admit, not often except in articles about cartoons or humor), 2. low-quality silly cartoons of this sort are often seen in math/science magazines accompanying articles similar to the ones this cartoon has been used in (was in Quintic equation, now in Bring radical). I seriously doubt that the presence of a cartoon would make many readers with a serious interest in a math article take the article less seriously, as the same group of people would likely have seen similar cartoons in similar contexts elsewhere. Also, so what if 99% of people in the world wouldn't get the pun; 99% of people in the world won't read the article. CyborgTosser (Only half the battle) 19:47, 20 March 2006 (UTC)
thread_title: Cartoon argument synopsis

message: Here's a copy of that discussion. Lunch 04:49, 15 April 2007 (UTC)
replies: text: Keep. It is funny. Besides, it does actually serve a pedagogical purpose in that it, on first reading, seems to contradict a known fact about solving quintics, but then after the pun is recognized, emphasizes distinction between types of solvability. --C S (Talk) 07:29, 18 March 2006 (UTC)

text: Delete, no fugly drawings on Wikipedia, please. I'd not take serious an encyclopedia article with that sort of illustration. Also not funny for those lacking an university education in math, i.e. 99% of the world. Sandstein 14:45, 18 March 2006 (UTC)

text: Keep. It is in use on someone's page, I see no harm in keeping it. Weatherman90 17:17, 18 March 2006 (UTC)
replies: text: Er, it says: "No pages link to here." Sandstein 18:29, 18 March 2006 (UTC)
replies: text: Check again. User:CyborgTosser
--Weatherman90 22:14, 18 March 2006 (UTC)
replies: text: Oh, now I see. I looked here. Still, I think it's just not done well enough to illustrate a scientific article. If you just want to keep it on your user page, no problem. Sandstein 07:17, 19 March 2006 (UTC)
replies: text: That was actually me who said "Check again", my signature got misplaced somehow. You must have read the userpage link that I pasted on there as a signature. The user who has it on his page has yet to comment. Weatherman90 16:04, 19 March 2006 (UTC)

text: Keep. To summarize my comments on Talk:Quintic equation, 1. silly cartoons are found in serious encyclopedias (although I admit, not often except in articles about cartoons or humor), 2. low-quality silly cartoons of this sort are often seen in math/science magazines accompanying articles similar to the ones this cartoon has been used in (was in Quintic equation, now in Bring radical). I seriously doubt that the presence of a cartoon would make many readers with a serious interest in a math article take the article less seriously, as the same group of people would likely have seen similar cartoons in similar contexts elsewhere. Also, so what if 99% of people in the world wouldn't get the pun; 99% of people in the world won't read the article. CyborgTosser (Only half the battle) 19:47, 20 March 2006 (UTC)
thread_title: IFD discussion from 18 March 2006

message: Where it says
:\left(-\frac{p}{4}\right)^\frac{1}{4}\operatorname{BR}\left(\frac{(-5/p)^\frac{5}{4} q}{4}\right),
I think it should be
:\left(-\frac{p}{5}\right)^\frac{1}{4}\operatorname{BR}\left(\frac{(-5/p)^\frac{5}{4} q}{-4}\right).
Can someone confirm that? When I plug the first form into x5+px+q, with the identity BR(k)5 = 5BR(k)+4k, I don't get zero. Black Carrot 12:28, 5 July 2007 (UTC)
replies: text: My calculations confirm yours. I have fixed the article. -- Meni Rosenfeld (talk) 10:35, 10 July 2007 (UTC)
thread_title: Bring Radical Formula

message: In the section giving the solution of the quartic equation in k, the Davis reference
Harold T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, 1962
on page 173 gives
sin(alpha) = 4/A^2 not sin(alpha) = 1/(4*A^2)
Derivation and numerical check also support Davis's expression for alpha.
Ouevres de Charles Hermite, Tome II, p. 11, also corrects this which was indeed present on page 513 of Hermite's original paper in Comptes Rendu, Vol 46, 1858(1), p. 508. Mathworld and the Mathematica poster give correct expressions, working out again to the correct one given above. There is another very obvious error in the middle of page 513 of Hermite's original paper, in which the first fraction in the expression for a shows the reciprocal (5^(5/4))/2 instead of the correct fraction 2/(5^(5/4)).
The quartic in k can be written as (1+k^2)^2=A^2*k*(1-k^2)
If one lets t=alpha/4, the solutions are claimed to be
k=tan(t), tan(t+pi/2), tan(pi/4-t), and tan(3*pi/4-t). To show this (for simplicity, we consider only the k=tan(t) solution)
use the subsidiary angle alpha as defined above. Translating the quartic above into trigonometric terms,
the A^2 term becomes 4/sin(alpha).
But sin(alpha)=sin(4t)=2sin(2t)cos(2t)=4sin(t)cos(t)[cos^2(t)-sin^2(t)]
1+k^2=1+tan^2(t)=sec^2(t) and 1-k^2 = 1-tan^2(t)=[cos^2(t)-sin^2(t)]/cos^2(t)
The left side of the equation becomes sec^4(t)= 1/cos^4(t).
The right side becomes
(4/(4sin(t)cos(t)[cos^2(t)-sin^2(t)])*(sin(t)/cos(t))*[cos^2(t)-sin^2(t)]/cos^2(t).
The 4's in numerator and denominator cancel, as do the sin(t)'s in numerator and denominator, as do the terms in square brackets, giving simply 1/cos^4(t) on the right side as well.
The validity of this identity shows that sin(alpha)=4/A^2 gives a solution of the quartic. The complete set of four solutions of the quartic are
k=tan(t), -1/k=tan(t+pi/2)=tan(t-pi/2), (1-k)/(1+k)=tan(pi/4-t)=-tan(t+3*pi/4), and (k+1)/(k-1)=tan(3*pi/4-t)=-tan(t+pi/4).
Any of the four solutions can also be expressed in similar fashion from any single valid solution, i.e. its negative reciprocal is another solution, etc.
The angle alpha obtained as the inverse sine of 1/(4*A^2) (as given in Hermite's original paper and in King's Beyond the Quartic Equation, p. 131)
does not give a valid trigonometric solution of the quartic, nor of course does it give correct values for k. If you want to check this numerically, you can try a simple value for k like 1/2, then find in turn alpha and A, and finally substitute in the quartic and check if the equation is satisfied.
Therefore I am again changing the expression for sin(alpha) back to 4/A^2
I note that the quartic is a quasi-recurrent quartic and could also be solved algebraically more easily than most quartics by dividing the quartic by k^2, then using u=k-1/k, u^2=k^2-2+1/k^2.
This gives u^2+A^2*u+4=0, a quadratic giving u on solution and then k as the solution of k^2-u*k-1=0.
However the trigonometric solution seems preferable.
Wvitale (talk) 14:09, 6 April 2009 (UTC)
thread_title: The Hermite-Kronecker-Brioschi characterization

message: I just edited the section titled Doyle–McMullen iteration. Before my edits, the section was titled Doyle-McMullen iteration, with a hyphen rather than a dash (see WP:MOS). Notice that when you used the "#" to generate a numbered list, it starts the numbering all over again from 1 if there is any interruption in the pattern, and that's what happened. Also, when a paragraph is indented, as by the "#" or by an initial bullet, or in any other way, then "displayed" TeX within the paragraph should be indented twice, and what comes after the TeX display will fail to be included within the bullet point or numbered item, and thus not indented, and thus appear not to be within the bullet point or numbered item, unless it's indented separately. Also, is there a reason to use a complicated mixture of TeX and html tables instead of using "align" within TeX? Michael Hardy (talk) 01:57, 28 May 2011 (UTC)
thread_title: Some editing

message: This paragraph suggested that –1 is a branch point of the Bring radical and that this is the unique branch point. This appears to be wrong, the branch points being the fourth roots of –1/5. I have thus made the paragraph left precise, but, I hope, correct.
By the way, has somebody considered the Bring radical for non-real a? If not, I suggest to remove the mention of the complex case. D.Lazard (talk) 20:28, 9 October 2015 (UTC)
thread_title: The second paragraph of the lead was wrong

message: \operatorname{BR}(a) looks like a function: \operatorname{BR}(a) = x such that x^5 + x + a = 0.
Someone must have done to the Bring radical function all of the things that are done to every other function:
*Show a table and/or graph of the function. (And then of its derivative, integral, etc.)
*Show expressions for \operatorname{BR}(a+b), \operatorname{BR}(ab), \operatorname{BR}(a^b), etc., or the impossibility thereof.
*Show infinite series for the function, or the impossibility thereof. (Including infinite sum, infinite product, extended fraction, infinite radical, etc.) (And then its derivative, integral, etc.)
*Show inverse function.
*Show iterated function.
*Show fractionally-iterated function...
— A876 (talk • contribs) 04:03, 25 October 2015‎
:I think this is a graph of BR(x), but please make sure of it before putting it in the article (if you decide to do that)—I am no mathematician! Also, note that you probably can't just take a screenshot of the graph and put that screenshot in the article as an image because doing that is likely a violation of Desmos's terms of service. If this is a correct graph, you should be able to find the approximate value of the Bring radical of a number by clicking on the graph of the function (the actual red curve) and moving your cursor until the red dot is at the x-value that you want. You could make a table this way. I don't think that the derivative or integral of BR(a) can be represented in a way that is finite and only uses addition, subtraction, multiplication, division, and exponentiation (including nth roots), considering that BR(a) itself can't be represented this way. Concerning infinite series, there is an infinite sum listed for -BR(a) in the "Series Representation" portion of the article. There's also this paper, which may provide additional infinite series.
:69.112.210.105 (talk) 03:21, 5 January 2018 (UTC)
thread_title: Would like to see the usual investigations

message: Hello fellow Wikipedians,
I have just added archive links to  on Bring radical. Please take a moment to review my edit. You may add  after the link to keep me from modifying it, if I keep adding bad data, but formatting bugs should be reported instead. Alternatively, you can add  to keep me off the page altogether, but should be used as a last resort. I made the following changes:
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thread_title:  External links modified 

message: I removed "irreducible" from in front of "polynomial" describing x^5 + x + a.  Presumably it was meant to mean irreducible over the rationals, but more importantly, this polynomial isn't irreducible for all values of a anyway, such as 0.
46.208.247.203 (talk) 20:01, 27 April 2016 (UTC)
thread_title:  Changed intro sentence 

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*Added archive https://web.archive.org/web/20090226035637/http://www.sigsam.org/bulletin/articles/145/Adamchik.pdf to http://www.sigsam.org/bulletin/articles/145/Adamchik.pdf
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thread_title:  External links modified 

message: Strangely, I did not find any note about Galois theory in this article. Really, missing? Shouldn't we tell the reader about Galois theory#A non-solvable quintic example? Boris Tsirelson (talk) 05:50, 9 February 2018 (UTC)
thread_title:  Not a usual radical 

message: In the Doyle–McMullen iteration section of this article, the equation for h(Z, w) should end with "- w^9" the current article has "+ w^9". 
See "Solving the Quintic by Iteration" page 33 here: https://math.dartmouth.edu/~doyle/docs/icos/icos.pdf
I think this equation is SVG graphics so I dont know how to make the fix.  — Preceding unsigned comment added by BartonFunk (talk • contribs) 20:58, 5 September 2019 (UTC)
thread_title:  There's an error in one of the equations in the Doyle-McMullen Iteration section