Thanks, David, that´s very helpful. I will look a bit deeper into these approaches.
Kind regards Achim David Roe schrieb am Freitag, 5. August 2022 um 23:11:54 UTC+2: > Hi Achim, > Many of the polynomials you mention can be factored by Sage if you use > number fields for your coefficients rather than the symbolic ring. For > example: > > sage: R.<x> = ZZ[] > sage: K.<w> = NumberField(x^2 + x + 1) > sage: f = x^5 + 9/2 * x^4 - 5/2 * x^3 - 2*w * x^2 - 9*w * x + 5*w > sage: f.factor() > (x - 1/2) * (x + 5) * (x^3 - 2*w) > > There's a separate question of trying to write the roots of an irreducible > polynomial in terms of radicals. The process for doing this depends on the > Galois group (you can find examples of number fields with each of the > possible > degree 5 Galois groups <https://beta.lmfdb.org/GaloisGroup/?n=5> using LMFDB > searches like this <https://beta.lmfdb.org/NumberField/?galois_group=5T1>). > If the Galois group is not solvable (S5 or A5 in the degree 5 case), it's > not possible to write roots in radicals. If it is solvable, you can find a > chain of subgroups where each successive quotient in the chain is cyclic, > and then use Kummer theory to express each extension as adjoining an nth > root. After expressing the Galois closure as an iterated extension in this > way, you can then factor your original polynomial in this field. > Computationally, this gets to be very expensive as the degree of the Galois > closure increases, but it's totally doable for quintics. > > If you want to learn more about this topic there are plenty of good > references on Galois theory. I think a function that used Sage's Galois > groups (which are computed by Pari under the hood) in order to express > roots of a polynomial symbolically in terms of nth roots (when possible) > would be a nice contribution. > David > > On Fri, Aug 5, 2022 at 1:44 PM Fat i <achi...@gmail.com> wrote: > >> Hello, >> >> I am new to this group and got the suggestion to post this here which I >> am happy to do. If you are interested in polynomials, esp. solving >> quintics, you may have a look at >> >> CoCalc -- Development >> <https://cocalc.com/Achim/SolvingQuintics/SolvingQuintics> >> >> I have spent some time studying quintics and implemented a class library >> which wraps and extends SAGE capabilities., Happy to receive your feedback >> or questions, or let me know if you would like to contribute or collaborate. >> >> Check the README.md file for an overview. >> >> Kind regards >> Achim >> >> -- >> You received this message because you are subscribed to the Google Groups >> "sage-devel" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to sage-devel+...@googlegroups.com. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/sage-devel/b5a63fbb-456b-446e-b270-abcefad57dabn%40googlegroups.com >> >> <https://groups.google.com/d/msgid/sage-devel/b5a63fbb-456b-446e-b270-abcefad57dabn%40googlegroups.com?utm_medium=email&utm_source=footer> >> . >> > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/63bede8b-8b91-413d-a645-994babb698ecn%40googlegroups.com.
