1

What are all polynomials of degree ≥ 5 that have solution formulas? Numerical methods like calculating for each number is not what I ask for.

The answer states:

Note that for some special cases (e.g., $x^n - a$), solution formulas exist, but they do not generalize to all polynomials. In fact, it is known that only a very small part of polynomials of degree $\ge 5$ admit a solution formula using the operations listed above.

The other answer in the same thread states:

There are general formulas for the general equations $x^n-x+t=0$ and $ax^{2\mu}+bx^\mu-x^\nu+c=0$

Community
  • 1
Ziya Ceferov
  • 141
  • 1
    xn-axn-a is unreadable. Please go to the help menu, and study up on how to format math on this website. – Gerry Myerson Mar 15 '16 at 11:42
  • @GerryMyerson, I feel sorry, but I added a link to the answer. I hope it is enough as I can't fix it right now, but I will try to fix it. – Ziya Ceferov Mar 15 '16 at 11:44
  • I don't want a link, I want a properly formatted question that I can read. Can you read xn-axn-a? How can you expect anyone else to. If worst comes to worst, write it out in words. – Gerry Myerson Mar 15 '16 at 11:45
  • Obviously, $ax^n+b=0$ is solvable, as is $ax^6+bx^3+c=0$ or $ax^6+bx^4+cx^2+d=0$ –  Mar 15 '16 at 11:47
  • @GerryMyerson, I can't fix now, because I don't know how to format, but I am already reading the reference to fix it. Is the question understandable? – Ziya Ceferov Mar 15 '16 at 11:48
  • I'll ask you again – can you understand xn-axn-a? If you can't, how do you expect me to? – Gerry Myerson Mar 15 '16 at 11:50
  • 1
    @GerryMyerson, I have fixed. I asked if the question itself understandable. Let me know if something is still wrong. – Ziya Ceferov Mar 15 '16 at 11:54
  • @YvesDaoust, do you mean numerical methods or solution formulas by saying "solvable"? Because any polynomial is solvable, but I need those that have solution formulas like those with degree $\le 4$, if you mean the latter, then I need a list of all polynomial or understand how to know which one has a solution formula and what is the solution formula. – Ziya Ceferov Mar 15 '16 at 12:03
  • I mean analytical solution. I don't think it is possible to establish an exhaustive list. –  Mar 15 '16 at 12:15
  • @Yves Daoust, I am not sure what exactly "analytical solution" means, but I guess it is solving by guessing options, i.e. neither of the 2 given options. I understand your last sentence as among many polynomials we can find ones with solution formulas, but to find them all we need to scan all the possible polynomials and we will be finding them as long as we scan them. I thought there are a few of them with solution formulas, but your idea is a bit different. By anyway, I got what mean. – Ziya Ceferov Mar 15 '16 at 12:29
  • 1
    @ZiyaCeferov: there are too many possibilities (infinitely many) to be able to give an exhaustive list. You need to read about Galois' theory. https://en.wikipedia.org/wiki/Galois_theory#Application_to_classical_problems –  Mar 15 '16 at 12:37
  • I don't believe there is a general formula for solving $x^n-x-t=0$ in radicals, not even for the case $n=5$. Dummit has written a paper on solving solvable quintics; it's at http://www.cem.uvm.edu/~ddummit/quintics/solvable.pdf – Gerry Myerson Mar 15 '16 at 21:48

0 Answers0