#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 | 09683145989c437b59e02c02409d1dac05ac379d
| Stopgaps:
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Comment (by gagern):
Replying to [comment:82 jdemeyer]:
> Can you please say where this example comes from?
I was looking for numbers with a minpoly of degree more than four which
allow for radical expressions but where computing said expression might
take some time. I intended to use this in a performance comparison. This
particular one I found using
{{{
sage: R.<x,y>=QQ[]
sage: (x^2 - 5*x + 2 - y).resultant(y^3 + y - 4, y)
-x^6 + 15*x^5 - 81*x^4 + 185*x^3 - 163*x^2 + 65*x - 6
sage: _.univariate_polynomial().roots(QQbar, False)
[0.1274914752251046?,
4.872508524774896?,
0.5703693585365878? - 0.4035749775931049?*I,
0.5703693585365878? + 0.4035749775931049?*I,
4.429630641463412? - 0.403574977593105?*I,
4.429630641463412? + 0.403574977593105?*I]
sage: _[1].minpoly()
x^6 - 15*x^5 + 81*x^4 - 185*x^3 + 163*x^2 - 65*x + 6
sage: K.<a> = NumberField(_, embedding=4.9)
}}}
I've essentially been toying with the original two polynomials until I got
something real and with reasonable conversion to symbolic. There is no
deeper meaning behind the actual values of the coefficients. The
conversion to symbolic is possible because the polynomial in `y` is cubic,
so one can express `y` using radicals, and once you know `y` the
polynomial in `x` is quadratic so that, too, can be expressed using
radicals. I was somewhat surprised that the symbolic expression could
avoid complex numbers along the way, since usually solving cubic equations
requires those.
I then did my performance test using `SR(a + k)` for some integer `k`. And
at some point decided that I should try more complicated elements of that
field as well, tried the `a + a^3` and was really surprised by the result.
--
Ticket URL: <http://trac.sagemath.org/ticket/14239#comment:83>
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