# Talk:Bring radical

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## Other Characterizations of the Bring Radical

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)

By the way, the derivation of the Bring radical as a power series in the main section is unreferenced, and might well be original research. I have never seen it anywhere except here on Wikipedia, and that's more than a little worrisome. Anyone know of any references which describe it? --Stormwyrm (talk) 11:00, 19 February 2009 (UTC)

## Extensions to higher order polynomials

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)

From what I know, the sextic equation and higher degree polynomial equations cannot be reduced into a single parameter form such as the Bring-Jerrard quintic form, and so their solution is somewhat more complicated. The technique I presented due to M.L. Glasser has been generalized to equations of arbitrarily high degree but using hypergeometric functions of several variables (see here for a German paper). The original quintic solution by Charles Hermite was later generalized to equations of arbitrary degree using Siegel modular forms. --Stormwyrm 03:45, 20 September 2006 (UTC)

## Radicals Disambiguation needed

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:

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)

## Cartoon

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Bring radicals can be used to obtain closed form solutions of quintic equations.

I am totally against including the cartoon in this article. It adds nothing to the article. It is unencyclopedic original artwork. This is the not the place for this. Wikipedia is not a publisher of original cartoons. --JW1805 (Talk) 04:26, 15 April 2007 (UTC)

JW, I see that you came to edit the article to do battle with Zephram Stark. I do appreciate your efforts to undo his damage. Thank you.
However, I respectfully ask that you refrain from further editing of this article. It seems to me that you have no mathematical training and do not understand the article (and the cartoon). You have not edited any mathematics or science articles other than this one, and your edit summaries and your comments above suggest you do not understand the pun in the cartoon.
The cartoon is not unencyclopedic. The pun brings home the point that quintics are not solvable with radicals: non-algebraic functions, Bring "radicals", are needed. Ned's friend is not asking Ned to bring (as in carry along) radicals, he is asking him to bring Bring radicals. Dissecting humor usually kills it, but I still think it is funny. (And I know I'm not alone in these opinions.) I might add that low-resolution ("lo fi") artwork is trendy.
Further, almost all artwork on Wikipedia is original artwork. It has to be since Wikipedia requires "free" images. This cartoon is no different. Lunch 04:49, 15 April 2007 (UTC)

I have again removed the cartoon. I like it, but we are trying to create a real encyclopedia here, and serious encyclopedias don't use cartoons to illustrate points. I'm sure we can use the cartoon in other wikimedia projects, such as wikiversity. nadav (talk) 00:45, 9 July 2007 (UTC)

And I have restored it. I disagree. --C S (Talk) 00:58, 9 July 2007 (UTC)
Can you explain why? nadav (talk) 01:14, 9 July 2007 (UTC)
I could, but I happen to agree with what has already been said by Lunch (above) and CyborgTosser (below). Also see my comment below from the previous IFD. If the rationales are not sufficiently detailed, just ask again and I will make an additional clarification on whatever was insufficiently explained. Since this is a mathematics article, I have asked WikiProject Mathematics for more opinions on this cartoon issue. --C S (Talk) 01:21, 9 July 2007 (UTC)
Addendum: I don't know why you really need a further explanation, as I think it obvious that this is something where people will just have different opinions on appropriateness. I don't buy your claim that Wikipedia is "serious" and thus cannot use cartoons to illustrate a point. Mathematical points are hard to get across oftentimes, and cartoons are sometimes a good way to make one. If you don't think the cartoon belong, ok. But I think it does. --C S (Talk) 01:27, 9 July 2007 (UTC)
Teachers use all kinds of pedagogic tools that do not belong in encyclopedias, and I think many Wikipedia editors would agree that cartoons shouldn't be used merely to humorously convey a point. If I cared enough about this, I would open an RfC, but I don't think it's worth my time. nadav (talk) 01:44, 9 July 2007 (UTC)
Pointing out that there are lots of other things that don't belong is a red herring. Of course I agree that lots of things, e.g. certain teaching tools, don't belong here. --C S (Talk) 01:55, 9 July 2007 (UTC)
It's not a red herring: if you agree that merely being a useful teaching tool is not enough to warrant inclusion in an encyclopedia, then you must find an additional reason for including the cartoon. nadav (talk) 02:26, 9 July 2007 (UTC)
No, you got it wrong; I don't need any additional reason other than it is useful. The reason some other things don't belong on Wikipedia is that they are contrary to WP:NOT. It's up to you to explain why this cartoon falls under that policy, not to me to give a reason beyond utility.
When something is useful for the reader, it should be included unless this runs counter to what Wikipedia is. That is the point of WP:NOT. Wikipedia is basically defined by what it is not, as there is nothing else that is like it. The only really pertinent section of WP:NOT is "Wikipedia is not a manual, guidebook, or textbook" yet I can't see how adding one cartoon makes it any of these things. I have voted to delete plenty of mathematical articles that fall under these categories, by the way. --C S (Talk) 02:43, 9 July 2007 (UTC)
Can you explain how the joke is "useful" for the stated purpose of the encyclopedia, to convey information? This image doesn't convey any information — in fact, it only makes sense if the reader already knows the information conveyed by the article. There is more to Wikipedia policy than WP:NOT, which only covers the most common misunderstandings for what Wikipedia is. Wikipedia is also not a pineapple, a life-saving device, a small child, or a place to display math puns — although you won't find these listed on WP:NOT. ~ Booya Bazooka 14:01, 10 July 2007 (UTC)
(unindent, response to Booya): Ouch, I hadn't considered that Wikipedia is not a small child! So my argument that this image is not policy or guideline violating has completely collapsed. I hadn't realized there were even policies against defining Wikipedia as a small child. Presumably this is contained in the policy that excludes math puns, although this is the first I've ever heard of this. As for "convey[ing] information". There is much on Wikipedia that strictly speaking, does not convey information in terms of some fact. There are materials that lend additional insight, make the learning experience more pleasurable, and work to reinforce material that is explained in another part of the article. I believe this image falls into this type of content. --C S (Talk) 15:08, 10 July 2007 (UTC)

(<--indent) I gave the reason for my opinion; you don't agree. Nothing more to discuss. And I never claimed to use WP:NOT; I was arguing based on personal taste and experience with other reference works (which I think Wikipedia should emulate). Continuing this argument is pointless. nadav (talk) 03:18, 9 July 2007 (UTC)

I thought it was clear from the start that this was a matter of taste, which is why I kept my first comment short and merely said I disagreed. You were the one that asked for further explanation! In my response to your request, I pointed out that I thought your reason was simply that you don't like having the image, while I do, making further discussion pointless. Then you continued the discussion by trying to poke a hole in something I said, so of course I will respond. To top it off, you even ended up demanding I give even more rationale based on what I had said! ("...you must find an additional reason for including the cartoon"), so of course I will respond to that too. --C S (Talk) 03:41, 9 July 2007 (UTC)
Delete. Although I can see both sides of the article, here are my reasons for removal. According to WP:NOT#TEXT, Wikipedia is not a textbook and "The purpose of Wikipedia is to present facts, not to teach subject matter." I agree that this image may help teach the material (hence the reason cartoons like this are often included in math articles), but I don't believe that this image presents or clarifies any facts regarding Bring Radicals. A person may remember and identify with Bring Radicals much easier having seen this cartoon, but they will not learn what Bring Radicals actually are any easier. -Weston.pace 18:10, 9 July 2007 (UTC)
I have found several low quality cartoons in an encyclopedia of mathematics that I came across(I can find you an ISBN if you like). I don't think that aspect of NOT really applies here. The cartoon will help them remember the whole article and the radical's relation to quintics IMO. --Cronholm144 10:58, 10 July 2007 (UTC)
See also the discussion at Wikipedia talk:WikiProject Mathematics#Cartoon at Bring radical. JRSpriggs 06:22, 10 July 2007 (UTC)
Thank you JRSpriggs, I will continue my discussion there. Weston.pace 14:00, 10 July 2007 (UTC)
Keep it. Many of our articles have a section "In popular culture", so we have established precedent for admitting (relevant) frivolity. As for censoring images, Wikipedia is firmly against it, even when some may consider the images provocative.
Some people take "serious" too seriously. Not the great ones. Richard Feynman, winner of the Nobel Prize in Physics, was famous for his sense of humor. And who is going to explain to Donald Knuth that serious encyclopedic works are prohibited from including cartoons and humor? No need to stop there; consider the serious journals and books containing humor chronicled by the University of Chicago's Crerar Library.
(But I'd lose the equations, vary the line weight along the length of a stroke for greater expressiveness, and use a nice cartoon font.) --KSmrqT 14:35, 10 July 2007 (UTC)
Done, but it is back to a png for rendering reasons. --Cronholm144 04:42, 11 July 2007 (UTC)
Maybe I do take things too seriously, I have read "Surely You're Joking, Mr. Feynman!" and "Further Adventures of a Curious Character", and I admire the man. He won a Nobel Prize after all, for writing brilliant physics like "Space-Time Approach to Non-Relativistic Quantum Mechanics" and "Theory of the Fermi Interaction." Might I point out that both those articles have no cartoons in them. Yes, there are also a number of serious journals and books containing humor, but those are the minority. I think humor is fantastic, but I don't believe it belongs in Wikipedia. Weston.pace 15:05, 10 July 2007 (UTC)
"I think humor is fantastic but I don't believe it belongs in Wikipedia." Wow, I guess this only confirms what I suspected. The dividing line is between the humor police and those with healthy senses of humor! ^_^. --C S (Talk) 15:13, 10 July 2007 (UTC)
Computer science majors are trained not to laugh. Weston.pace 15:22, 10 July 2007 (UTC)
Last time I laughed was in grade school, and I've regretted it ever since. nadav (talk) 04:14, 11 July 2007 (UTC)

Hello, I have no expierence with either the article, the topic matter, or any of the editors in this discussion. My expierence in math is limited to early (AB) calculus, I came here because of a request for more discussion at Wikipedia:Village_pump_(policy)/Archive AT#Inclusion_of_jokes. To me it seems that the image should go. I don't think it's encyclopedic in any way, and its inclusion is not necessary to understand the concept. As stated before, Wikipedia is not made to teach subject matter. It looks really unproffesional, and while most mathematics related articles include images to visualize a geometric or graphing concept, this is related more to language than it is to math. --YbborTalk 13:49, 16 July 2007 (UTC)

I say keep it. Admittedly, WP:ILIKEIT, but if we're supposed to learn about "bring radicals", the cartoon is an aid to memory, and such tricks are common in college textbooks. Shalom Hello 21:39, 16 July 2007 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

## Cartoon argument synopsis

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)

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)
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)

I don't think there is an exact policy. Though they are discussing it at the village pump(see above).--Cronholm144 14:04, 16 July 2007 (UTC)

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)
• I don't see a problem with it, and in a way I think that it brings some closure to those who have the obvious joke in mind and once it's seen they move past the thought. No harm and why not have a bit of silly fun. --Kevin Murray 22:28, 16 July 2007 (UTC)

As someone who has no idea what a bring radical is (having not read the article for the explicit purpose of providing a know-nothing opinion), when I first looked at the cartoon I learned that "Bring radicals can be used to solve quintic equations." This other stuff I hear about radicals being useless for solving quintic equations on this talk page is not included in the cartoon, and it doesn't reinforce the point at all.

Now I've read the article. The caption, basically, is just as useful as the cartoon without making a feeble and unencyclopedic joke. Mathematics and science articles are often dense and less useful to an outsider, but this is not the way to solve that issue; pun cartoons do nothing but make the article seem unprofessional. Atropos 23:11, 18 July 2007 (UTC)

• It's actually a bad cartoon, in that it's neither informative nor funny. The caption on the other hand, is useful; the cartoon's primary purpose here appears to be to attract attention to the caption. >Radiant< 12:35, 6 August 2007 (UTC)
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)
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)
• I believe we've reached consensus, which is I why I reverted User:Lunch's revert. This topic had a debate on IFD (summarized above) which didn't reach enough consensus to remove it, then there was a debate on this talk page (archived above) which didn't reach consensus, it moved to a debate on the Math project talk page which began to reach consensus, it also went to the village pump, the arguments were then summarized above and brought here, where it seems to be in consensus. Through all this discussion the consensus seems to be that it doesn't belong, but no one is going to fight strongly for it's removal. Someone removed it, I think we should leave it that way. -Weston.pace 20:40, 17 August 2007 (UTC)

I disagree; I don't see any concensus for removal. At least six long-time editors of math articles have made several arguments above for keeping the cartoon. (I count myself, C-S, Cronholm144, JRSpriggs, KSmrq, and CyborgTosser; if you look through the discussions on WT:WPM, there are more.) I don't see a real rebuttal for these arguments. On the other hand, the people arguing for the cartoon's deletion have never edited this article (outside the cartoon issue); most have never even edited a math article. At least you, Weston, have that going for you, but I think you should stick around longer to get a better feel for the style employed in math articles.

Since Radiant hasn't deigned to tell anyone, the cartoon was actually nominated for IFD, and was kept. It is now at deletion review. (And NB: the cartoon was nominated for deletion before, too; it was kept then as well.) Sigh. Lunch 20:51, 17 August 2007 (UTC)

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)
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)
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)

### IFD discussion from 18 March 2006

Here's a copy of that discussion. Lunch 04:49, 15 April 2007 (UTC)

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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)
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)
Keep. It is in use on someone's page, I see no harm in keeping it. Weatherman90 17:17, 18 March 2006 (UTC)
Er, it says: "No pages link to here." Sandstein 18:29, 18 March 2006 (UTC)
Check again. User:CyborgTosser

--Weatherman90 22:14, 18 March 2006 (UTC)

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)
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)
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)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

## Bring Radical Formula

Where it says

${\displaystyle \left(-{\frac {p}{4}}\right)^{\frac {1}{4}}\operatorname {BR} \left({\frac {(-5/p)^{\frac {5}{4}}q}{4}}\right)}$,

I think it should be

${\displaystyle \left(-{\frac {p}{5}}\right)^{\frac {1}{4}}\operatorname {BR} \left({\frac {(-5/p)^{\frac {5}{4}}q}{-4}}\right)}$.

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)

My calculations confirm yours. I have fixed the article. -- Meni Rosenfeld (talk) 10:35, 10 July 2007 (UTC)

## The Hermite-Kronecker-Brioschi characterization

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)

I stand corrected. I will however leave a note in the article that some important references are in error. --Stormwyrm (talk) 05:14, 8 April 2009 (UTC)

## Some editing

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)

## The second paragraph of the lead was wrong

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)

## Would like to see the usual investigations

${\displaystyle \operatorname {BR} (a)}$ looks like a function: ${\displaystyle \operatorname {BR} (a)=x}$ such that ${\displaystyle x^{5}+x+a=0.}$

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 ${\displaystyle \operatorname {BR} (a+b)}$, ${\displaystyle \operatorname {BR} (ab)}$, ${\displaystyle \operatorname {BR} (a^{b})}$, 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 (talkcontribs) 04:03, 25 October 2015‎

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## Changed intro sentence

I removed "irreducible" from in front of "polynomial" describing ${\displaystyle x^{5}+x+a}$. 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)

## I dont know what to call this

I had this crazy idea to find BR(x). So, the Bring Radical of a is:

${\displaystyle x^{5}+x+a=0}$

If we make the substitution

${\displaystyle x=a*tan^{2}(y)}$

then we get

${\displaystyle a^{5}*tan^{1}0(y)+a*tan^{2}(y)+a=0}$

Note: the stray 0 is supposed to be in the exponent as tan^10. For some reason it wont go in the exponent with the 1.

We can group the last two terms together:

${\displaystyle a^{5}*tan^{1}0(y)+a*(tan^{2}(y)+1)=0}$

And use the secant-tangent identify:

${\displaystyle a^{5}*tan^{1}0(y)+a*sec^{2}(y)=0}$

And then change them into sines and cosines:

${\displaystyle a^{5}*sin^{1}0(y)/cos^{1}0(y)+a/cos^{2}(y)=0}$

Divide by ${\displaystyle a}$,

${\displaystyle a^{4}*sin^{1}0(y)/cos^{1}0(y)+1/cos^{2}(y)=0}$

Multiply out the denominators (multiply the entire equation to get rid of denominators, in this case by ${\displaystyle cos^{1}0(y)}$),

${\displaystyle a^{4}*sin^{1}0(y)+cos^{8}(y)=0}$

This is the sum of squares and is factorable over the complex plane:

${\displaystyle (a^{2}*sin^{5}(y)+cos^{4}(y)i)*(a^{2}*sin^{5}(y)-cos^{4}(y)i)=0}$

Therefore:

${\displaystyle a^{2}*sin^{5}(y)+cos^{4}(y)i=0}$ or ${\displaystyle a^{2}*sin^{5}(y)-cos^{4}(y)i=0}$

Divide by ${\displaystyle cos^{4}(y)}$,

${\displaystyle a^{2}*sin(y)*tan^{4}(y)=i}$ or ${\displaystyle a^{2}*sin(y)*tan^{4}(y)=-i}$

Note: I will add more at a later date. I do find it fascinating that a trigonometric substitution can factor the equation of a Bring Radical, with some work.

32ieww (talk) 18:46, 24 June 2016 (UTC)

This page is devoted to discuss how improving the article, not for blogging about original research on the subject. Nevertheless, for this time, I'll give an answer.
Your substitution ${\displaystyle x=a\tan ^{2}(y)}$ may be decomposed into ${\displaystyle x=az}$ and ${\displaystyle z=\tan ^{2}(y).}$ So, for each solution for x and z you introduce two opposite solutions for tan(y) (the two square roots of z). This does not induces a factorization for any equation, but it is the beginning of an explanation for the factorization that you have found. In any case, each factor of your factorization gives one square root of x/a for each root of the original equation, and thus only one (any) of the factors of your factorization must be kept: the other gives the same values for x. Also note that the substitution ${\displaystyle x=-a\sin ^{2}(y)}$ leads to ${\displaystyle a^{4}\sin ^{10}(y)=\cos ^{2}(y),}$ and thus to a simpler factorization. D.Lazard (talk) 17:37, 25 June 2016 (UTC)
So then we have

${\displaystyle (a^{2}\sin ^{5}(y)+cos(y))*(a^{2}\sin ^{5}(y)-cos(y))=0}$

Divide by ${\displaystyle a^{2}*cos(y)}$,

${\displaystyle sin^{5}(y)/cos(y)=1/a^{2}}$

${\displaystyle sin^{5}(y)/cos(y)=-1/a^{2}}$

Simplify:

${\displaystyle 10\sin(y)-5\sin(3y)+sin(5y)=16\cos(y)/a^{2}}$

${\displaystyle 10\sin(y)-5\sin(3y)+sin(5y)=-16\cos(y)/a^{2}}$

Any ideas, D.Lazard? 32ieww (talk) 14:33, 26 June 2016 (UTC)

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