Hello all, with the described class library, it is now possible to solve irreducible solvable Bring-Jerrard quintics, i.e. f(x) = x^5 + ax + b. Coefficients are calculated up to a certain limit which is based on the Cantor counting scheme of rational numbers with default maxValue = 20. Higher number coefficients max be generated if needed; however, due to O(n^4) complexity in generating the Spearman Williams coefficients, this is limited to maxValue = 100 currently. Contact the author for an extension if higher number coefficients are required.
I have copied over the updated files, including a test script called TestWorksheetBJ.sagews. Any feedback is appreciated. It might be interesting also to look at quartic Tschirnhaus transformations of general quintics which yield solvable Bring-Jerrard quintics. Best regards Achim Fat i schrieb am Samstag, 6. August 2022 um 08:40:59 UTC+2: > 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/4ba0045d-0054-44ef-a8ea-713d6e4efc0fn%40googlegroups.com.
