SymPy has algorithms to find roots of quintics in radicals (when they
exist). I don't recall if the algorithms work for symbolic inputs.
One can take a general quintic (x**5 + a*x**4 + b*x**3 + c*x**2 + d*x + e)
and shift it by y (replace x with x - y). Then expand and collect terms in
x. The coefficient of x**3 is a quadratic in y. Hence, one can solve a
quadratic in y in radicals terms of a and b, and shift the quintic by that.
One then has a new quintic, with no cubic term, with the same roots shifted
by some term which is expressible in radicals. Hence, the general quintic
with no cubic term is not solvable in radicals, as a solution would give a
solution to the general quintic (shift it back by the radical expression
above, which would keep it in radicals).
Here is some SymPy code:
a, b, c, d, e, x, y = symbols('a b c d e x y')
q = x**5 + a*x**4 + b*x**3 + c*x**2 + d*x + e
print(collect(q.subs(x, x - y).expand(), x, evaluate=False)[x**3])
t = solve(collect(q.subs(x, x - y).expand(), x, evaluate=False)[x**3], y)[0]
print(t)
print(collect(q.subs(x, x - t).expand(), x, evaluate=False).get(x**3))
Aaron Meurer
On Wed, Jan 20, 2016 at 4:55 AM, Oscar Benjamin <[email protected]>
wrote:
> On 20 January 2016 at 05:46, Denis Akhiyarov <[email protected]>
> wrote:
> > On Tuesday, January 19, 2016 at 11:41:47 PM UTC-6, Denis Akhiyarov wrote:
> >>
> >> no algebraic roots according to this theorem:
> >> https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
>
> The theorem only shows that a general algebraic solution for *all*
> quintics (or higher degree polynomials) is not possible. In this case
> it is not a fully general quintic since the coefficients of x^3 and
> x^2 are both zero. I'm not sure how to check based on the coefficients
> of a polynomial whether or not its Galois group is solvable. Can sympy
> do that?
>
> To the OP: do you need to solve this in terms of symbols A, B etc. or
> is it acceptable to solve it using particular numbers for the
> coefficients? You may have better luck using the actual numbers.
>
> > actually this case looks like has some special properties and hence has
> some
> > roots according to Wolfram:
> >
> > http://www.wolframalpha.com/input/?i=A*x%5E5%2BB*x%5E4%2BC*x-D%3D0
>
> My interpretation of that Wolfram output is that Wolfram is unable to
> solve this quintic (or rather this general family of quintics).
>
> --
> Oscar
>
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