I missed A=0 on wolfram, sorry for confusion On Wed, Jan 20, 2016, 9:38 AM Aaron Meurer <[email protected]> wrote:
> On Wed, Jan 20, 2016 at 10:30 AM, Oscar Benjamin < > [email protected]> wrote: > >> On 20 January 2016 at 15:11, Aaron Meurer <[email protected]> wrote: >> > 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). >> >> The above argument extends to making any of the coefficients zero with >> the exception of the constant coefficient e. If we could shift to x - >> t so that we have a new polynomial with zero constant term then x = t >> would be a root of the original polynomial which would imply being >> able to find one root of (and hence all roots of) any quintic. >> Otherwise though we can shift to make any of a, b, c or d zero. >> >> However the OP asked about a less general case where the cubic and >> quadratic coefficients are both zero. It's not possible in general to >> shift a quintic so that both b and c are zero. >> > > Oh I didn't notice that :) > > >> > 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)) >> >> To make b and c zero we'd need to solve these two equations >> simultaneously for one variable y: >> >> In [5]: print(collect(q.subs(x, x - y).expand(), x, evaluate=False)[x**2]) >> 6*a*y**2 - 3*b*y + c - 10*y**3 >> >> In [6]: print(collect(q.subs(x, x - y).expand(), x, evaluate=False)[x**3]) >> -4*a*y + b + 10*y**2 >> >> Clearly that will only work for some lucky values of a, b, and c. (One >> that jumps out is the trivial solution that we could have y=0 if b and >> c are both 0.) >> >> So this is not a general quintic but neither sympy nor Wolfram can >> solve it. I was wondering if it is possible to specifically verify >> that it has an unsolvable Galois group but perhaps not. >> > > SymPy is not able to do that (I checked and the roots_quintic function in > the polys only works for rational coefficients). > > If WolframAlpha can compute Galois groups I wasn't able to get it to work. > Probably GAP can do it, if someone wants to check that. > > Aaron Meurer > > >> >> -- >> Oscar >> >> -- > > >> You received this message because you are subscribed to the Google Groups >> "sympy" group. >> > To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected]. > > >> To post to this group, send email to [email protected]. >> Visit this group at https://groups.google.com/group/sympy. >> > To view this discussion on the web visit >> https://groups.google.com/d/msgid/sympy/CAHVvXxSFaF4BPbg0uhPMT2278NmRrmKj%2BUJsw4m04vOPTzvrQQ%40mail.gmail.com >> . >> For more options, visit https://groups.google.com/d/optout. >> > -- > You received this message because you are subscribed to a topic in the > Google Groups "sympy" group. > To unsubscribe from this topic, visit > https://groups.google.com/d/topic/sympy/sFNgH5zRNm0/unsubscribe. > To unsubscribe from this group and all its topics, send an email to > [email protected]. > To post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/sympy. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/CAKgW%3D6K3hzRQp3qNbVj5RkbWfZ03YrzRid1ck18WNz5JzpvW0A%40mail.gmail.com > <https://groups.google.com/d/msgid/sympy/CAKgW%3D6K3hzRQp3qNbVj5RkbWfZ03YrzRid1ck18WNz5JzpvW0A%40mail.gmail.com?utm_medium=email&utm_source=footer> > . > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/CALxxJLSTc3NW9L3gLLaqvNjAUsD4WgpNcBZp_20%3D_zJocFb99A%40mail.gmail.com. For more options, visit https://groups.google.com/d/optout.
