Solving quintics

[Fat i]

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

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

[David Roe]

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 using LMFDB searches like this).  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

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[Fat i]

Thanks, David, that´s very helpful. I will look a bit deeper into these approaches.

Kind regards

Achim

[Fat i]

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

[kcrisman]

For reference, is there a Trac ticket for this?  (But don't look now, wait until the current outage is resolved!)