If you have algebraic numbers $x$ and $y$, and you know the polynomials of least degree of which each is a solution (written as a vectors of coefficients), then how is the vector of coefficients of the polynomial of least degree of which $x y$ is a solution found; and likewise for $x+y$?

  • $\begingroup$ Absolutely not! I'm asking for a recipe for explicitly computing the vector of coefficients of the polynomial is a solution - not for a proof that it exists. $\endgroup$ – AmbretteOrrisey Dec 14 '18 at 9:05
  • $\begingroup$ @Hans Lundmark -- However ... now that I look at the answers more carefully, I see that there is one that might possibly give the solution. So thankyou for signposting it for me. I'm not sure it does give the answer yet ... but it kindof looks like it might. $\endgroup$ – AmbretteOrrisey Dec 14 '18 at 9:07
  • $\begingroup$ The accepted answer definitely says how to do it. (You have to look up how to compute resultants, and the formulas that you get aren't going to be simple, but that's just how it is.) $\endgroup$ – Hans Lundmark Dec 14 '18 at 9:12
  • $\begingroup$ And anyway the question is a duplicate, since that other question also asks for an explicit formula for a polynomial having $xy$ or $x+y$ as a root. $\endgroup$ – Hans Lundmark Dec 14 '18 at 9:14
  • $\begingroup$ @Hans Lundmark -- I do think I might be able to extract a solution (or maybe even more than one!) from it. It's extremely interesting post! But I could do with something abitt more explicit & specifically tuned to my question. Like ... what's this resultant? But at least that gives me a handle on it that I didn't previously have. $\endgroup$ – AmbretteOrrisey Dec 14 '18 at 9:16