symbolic radical expression for algebraic number

[gagern]

It would be nice to have a function which converts algebraic numbers to symbolic expressions using radicals, at least for those cases where this is possible (i.e. the cases where the minimal polynomial of the algebraic number has a degree less than 5). For the quadratic case, I have a code snippet to illustrate what I'm asking for:

def AA2SR(x):
    x = AA(x)
    p = x.minpoly()
    if p.degree() < 2:
        return SR(QQ(x))
    if p.degree() > 2:
        raise TypeError("Cannot handle degrees > 2")
    c, b, a = p.coeffs()
    y = (-b+sqrt(b*b-4*a*c))/(2*a)
    if x == AA(y):
        return y
    y = (-b-sqrt(b*b-4*a*c))/(2*a)
    assert x == AA(y)
    return y

def QQbar2SR(x):
    x = QQbar(x)
    return AA2SR(x.real()) + I*AA2SR(x.imag())

These functions could then be applied to vectors or matrices over algebraic real or complex numbers in order to obtain an exact description of the relevant values using radicals.

This request is a spin-off from my question on Ask Sage.

Follow-up tickets: #17516, #17517

Depends on #17495
Depends on #16964

Component: number fields

Author: Martin von Gagern, Jeroen Demeyer

Branch/Commit: 4626286

Reviewer: Marc Mezzarobba, Jeroen Demeyer, Vincent Delecroix

Issue created by migration from https://trac.sagemath.org/ticket/14239

[gagern]

Attachment: trac_14239_algebraic_to_radicals.patch.gz

[gagern]

comment:1

I wrapped my code into two methods radical_expression for the AlgebraicNumber and AlgebraicReal classes.

Only second degree polynomials supported so far. I guess this should cover the most useful use cases: third and fourth degree are certainly possible, but don't help very much with the readability of the result in most cases. Square roots, on the other hand, are everywhere, so being able to represent these is a huge win imho.

The code I wrote does rely little on the internal workings of algebraic numbers and their descriptions, mostly because I haven't dug into that code yet. I guess the check which solution of the quadratic equation to choose might benefit from a more direct comparison of separating intervals, but at the moment I see no urgent performance issue with most use cases of this function.

[mezzarobba]

Reviewer: Marc Mezzarobba

[mezzarobba]

comment:3

I think the feature you are looking for sort of exists. But it is available for elements of number fields, not algebraic numbers. And it is pretty inconvenient to use with algebraic numbers because alg.as_number_field_element() returns alg as an element of an abstract number field (plus an embedding of that number field into QQbar or AA), while conversion to symbolic expressions expects a number field with a registered embedding into CC or RR.

Yet, with the example from your question on AskSage, one can do:

sage: a = N[0][0]
sage: nf, val, nf2alg = a.as_number_field_element()
sage: approx = CC(nf2alg(nf.gen()))
sage: embedded_nf = NumberField(nf.polynomial(), 'a', embedding=approx)
sage: nf2nf = sage.rings.number_field.number_field_morphisms.NumberFieldEmbedding(nf, embedded_nf, embedded_nf.gen())
sage: SR(nf2nf(val))
1/5*sqrt(5)

So I disagree with the approach you take in your patch: IMO, we should make something like the above example work automatically and share the actual conversion code between algebraic numbers and number fields instead of implementing essentially the same feature twice.

[edit: add missing word]