#14239: symbolic radical expression for algebraic number
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Reporter: gagern | Owner: davidloeffler
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.4
Component: number fields | Resolution:
Keywords: | Merged in:
Authors: Martin von Gagern | Reviewers: Marc Mezzarobba,
Report Upstream: N/A | Jeroen Demeyer
Branch: | Work issues:
u/gagern/ticket/14239 | Commit:
Dependencies: #17495, #16964 | a5983f73b2acdbdd197f25d07c1fcf14bf1109e5
| Stopgaps:
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Comment (by gagern):
Looking back to comment:3 I find that for the example from comment:67 that
approach is able to find a radical expression on the level of the
algebraic numbers, without knowledge of the original number field:
{{{
sage: c = AA.polynomial_root(b.minpoly(), RIF(120,121))
sage: nf, val, nf2alg = c.as_number_field_element()
sage: val.polynomial()(nf2alg(nf.gen()).radical_expression())
1/8*((sqrt(4*(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) -
4/3/(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) + 17) +
13)*(sqrt(4*(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) - 4/3/(1/9*sqrt(109)*sqrt(3)
+ 2)^(1/3) + 17) + 1) + 52)*(sqrt(4*(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) -
4/3/(1/9*sqrt(109)*sqrt(3) + 2)^(1/3) + 17) + 1) + 10
}}}
So in a certain sense, `as_number_field_element` makes it easier to find a
symbolic expression for a given algebraic number, even though the
expression found in this way may be more complicated. I wonder whether I
should include that into my `radical_expression` method. Probably with a
switch to allow disabling it.
--
Ticket URL: <http://trac.sagemath.org/ticket/14239#comment:71>
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