8
$\begingroup$

In Ramanujan's Notebooks Volume IV pg. 31 by Bruce C. Berndt, he describes an easy way to solve the general quartic by starting with the system$$x^2+ay=b\\y^2+cx=d\tag1$$ And solving for $x$; which gives you the depressed form of the quartic.

Now, I'm wondering whether Ramanujan had his own methods for solving the general cubic, quintic, etc.

And according to this MSE post, Ramanujan had his own method for solving a solvable quintics.

Question: What was Ramanujan's method for solving the solvable quintic and quartic?

Note: I mean the sort of sneaky system (like $(1)$) employed by Ramanujan, and some substitution.

$\endgroup$
  • 1
    $\begingroup$ Could you please clarify the meaning of the word "solving"? I am asking bacuse it is impossible to solve the general quintic in radicals. $\endgroup$ – SashaP Oct 30 '16 at 19:36
  • $\begingroup$ According to your own reference in Does anyone know about Ramanujan's method of solving the quartic?, "Ramanujan devised his own method of solving the Quartic Equation after he learnt to solve the Cubic Equation", so the method for solving the cubic was not his own. As for the quintic and higher, Abel proved that there is no method for solving those by algebraic manipulations at all. $\endgroup$ – Conifold Oct 30 '16 at 21:08
  • $\begingroup$ @SashaP Hm.. true. Perhaps I should add the most important term "solvable" in there... $\endgroup$ – Frank Oct 30 '16 at 22:44
  • $\begingroup$ @Conifold Think of this question as an extension for the linked one... :) $\endgroup$ – Frank Oct 31 '16 at 5:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.