# Thread: Is the general polynomial solved?

1. ## Is the general polynomial solved?

My science fair project is concerned with solving a general polynomial of arbitrary degree. Naturally, I've wanted it to be completely original, but I've found several citations regarding its already having been done:

Polynomial - Wikipedia, the free encyclopedia
Solving the Quintic (the section titled "solution based on series")
Abel's Impossibility Theorem@Everything2.com (mentioned at the very bottom of the article)

You can see that the wikipedia source is without citation for the part that mentions hypergeometric functions.

My questions is: really?! Part of my project is considering the implications of solving the problem, and they seem to be momentously profound. I'm talking prime number, Riemann hypothesis profound, so why have I been able to find only three sources and a research paper that I have to buy (click here to see what I'm talking about--it's the article that is mentioned in the 4th note of the above wikipedia article) on the subject? I'm currently studying linear algebra so I don't really know what an eigenvalue is, but I've looked up the motivation for the hypergeometric function and its application to finding zeros has at least held intuitively for me, and it may have some clear application. But apparently it's been shown to have a direct application! Gahh!!

Dfrtbx

I guess that there's actually two questions in there... I actually do want both to be answered =)

2. Originally Posted by Dfrtbx
My science fair project is concerned with solving a general polynomial of arbitrary degree. Naturally, I've wanted it to be completely original, but I've found several citations regarding its already having been done:

Polynomial - Wikipedia, the free encyclopedia
Solving the Quintic (the section above the colorful picture)
Abel's Impossibility Theorem@Everything2.com (mentioned at the very bottom of the article)

You can see that the wikipedia source is without citation for the part that mentions hypergeometric functions.

My questions is: really?! Part of my project is considering the implications of solving the problem, and they seem to be momentously profound. I'm talking prime number, Riemann hypothesis profound, so why have I been able to find only three sources and a research paper that I have to buy (click here to see what I'm talking about--it's the article that is mentioned in the 4th note of the above wikipedia article) on the subject? I'm currently studying linear algebra so I don't really know what an eigenvalue is, but I've looked up the motivation for the hypergeometric function and its application to finding zeros has at least held intuitively for me, and it may have some clear application. But apparently it's been shown to have a direct application! Gahh!!

Dfrtbx

I guess that there's actually two questions in there... I actually do want both to be answered =)

I can't even see one single, poor question...
What are you asking EXACTLY?

Tonio

3. Originally Posted by Dfrtbx
My science fair project is concerned with solving a general polynomial of arbitrary degree. Naturally, I've wanted it to be completely original, but I've found several citations regarding its already having been done:

Polynomial - Wikipedia, the free encyclopedia
Solving the Quintic (the section above the colorful picture)
Abel's Impossibility Theorem@Everything2.com (mentioned at the very bottom of the article)

You can see that the wikipedia source is without citation for the part that mentions hypergeometric functions.

My questions is: really?! Part of my project is considering the implications of solving the problem, and they seem to be momentously profound. I'm talking prime number, Riemann hypothesis profound, so why have I been able to find only three sources and a research paper that I have to buy (click here to see what I'm talking about--it's the article that is mentioned in the 4th note of the above wikipedia article) on the subject? I'm currently studying linear algebra so I don't really know what an eigenvalue is, but I've looked up the motivation for the hypergeometric function and its application to finding zeros has at least held intuitively for me, and it may have some clear application. But apparently it's been shown to have a direct application! Gahh!!

Dfrtbx

I guess that there's actually two questions in there... I actually do want both to be answered =)
It has been proved that it's impossible to solve the general polynomial of degree 5 or greater algebraically. Contrary to your research, there is no scarcity of the proof. The proof will be found in any textbook that covers Field Theory. There is also an accessible account by Mario Livio (The Equation That Couldn't be Solved).

4. Originally Posted by mr fantastic
It has been proved that it's impossible to solve the general polynomial of degree 5 or greater algebraically. Contrary to your research, there is no scarcity of the proof. The proof will be found in any textbook that covers Field Theory. There is also an accessible account by Mario Livio (The Equation That Couldn't be Solved).
Yes: Abel. I think we may be on different pages here... I know about the proof about the impossibility of an algebraic solution. I'm concerned with a non-algebraic solution, and that is what is the citations are about--a non-algebraic solution. The article at everything2 says: "Felix Klein developed in 1877 a particularly elegant method for solving quintics based on the symmetries of the icosahedron." It says that he found a solution to every (I assume, it doesn't say so explicitly) quintic. It says in the very last paragraph that "Poincare among others" have extended Klein's approach to every polynomial of arbitrary degree.

Let me give you my motivation for a solution in terms of non-algebraic functions (this wasn't the initial motivation, but it's the one that makes the most sense): By the fundamental theorem of algebra, there must be n roots of an nth degree polynomial, and given integer coefficients, these roots must be algebraic. The Abel-Ruffini theorem concerns the impossibility of using algebraic functions in doing so--but there must be roots and these roots must be algebraic. I've taken this as a tell that non-algebraic functions can be used to find the roots.

Thanks for the book suggestion!

I can't even see one single, poor question...
What are you asking EXACTLY?

Tonio
Haha, that made me laugh... My english teachers have never liked the way I write, and probably with good reason =P

"I'm talking prime number, Riemann hypothesis profound, so why have I been able to find only three sources and a research paper that I have to buy ... on the subject?"

To be more explicit: has it really been done? And if it really has been done, why is the evidence of its having been done so scarce? It doesn't seem like it should be difficult to find information...

Dfrtbx

5. Originally Posted by Dfrtbx
Yes: Abel. I think we may be on different pages here... I know about the proof about the impossibility of an algebraic solution. I'm concerned with a non-algebraic solution, and that is what is the citations are about--a non-algebraic solution. The article at everything2 says: "Felix Klein developed in 1877 a particularly elegant method for solving quintics based on the symmetries of the icosahedron." It says that he found a solution to every (I assume, it doesn't say so explicitly) quintic. It says in the very last paragraph that "Poincare among others" have extended Klein's approach to every polynomial of arbitrary degree.

Let me give you my motivation for a solution in terms of non-algebraic functions (this wasn't the initial motivation, but it's the one that makes the most sense): By the fundamental theorem of algebra, there must be n roots of an nth degree polynomial, and given integer coefficients, these roots must be algebraic. The Abel-Ruffini theorem concerns the impossibility of using algebraic functions in doing so--but there must be roots and these roots must be algebraic. I've taken this as a tell that non-algebraic functions can be used to find the roots.

[snip]
No.

Sorry, but I think you're quite out of your depth here.

6. Originally Posted by mr fantastic
No.