>From wikipedia: Finding the roots of a given [quintic] polynomial has been a prominent mathematical problem.
Solving linear <http://en.wikipedia.org/wiki/Linear_equation>, quadratic<http://en.wikipedia.org/wiki/Quadratic_equation>, cubic <http://en.wikipedia.org/wiki/Cubic_equation> and quartic equations<http://en.wikipedia.org/wiki/Quartic_equation>by factorization <http://en.wikipedia.org/wiki/Factorization> into radicals<http://en.wikipedia.org/wiki/Nth_root>is fairly straightforward, no matter whether the roots are rational or irrational, real or complex; there are also formulae that yield the required solutions. However, there is no formula for general quintic equations over the rationals in terms of radicals; this is known as the Abel–Ruffini theorem <http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem>, first published in 1824, which was one of the first applications of group theory<http://en.wikipedia.org/wiki/Group_theory>in algebra. This result also holds for equations of higher degrees. An example quintic whose roots cannot be expressed by radicals is [image: x^5 - x + 1 = 0.] This quintic is in Bring–Jerrard normal form<http://en.wikipedia.org/wiki/Bring%E2%80%93Jerrard_normal_form> . As a practical matter, exact analytic solutions for polynomial equations are often unnecessary, and so numerical methods such as Laguerre's method<http://en.wikipedia.org/wiki/Laguerre%27s_method>or the Jenkins-Traub method <http://en.wikipedia.org/wiki/Jenkins-Traub_method> are probably the best way of obtaining solutions to general quintics and higher degree polynomial equations that arise in practice. However, analytic solutions are sometimes useful for certain applications, and many mathematicians have tried to develop them. Jason moorepants.info +01 530-601-9791 On Wed, Jun 5, 2013 at 8:23 AM, Stefan Krastanov <[email protected] > wrote: > I checked it with Mathematica. It gives the same answer. > > > On 5 June 2013 16:41, Aaron Meurer <[email protected]> wrote: > >> Prasoon recently implemented a complete algorithm for solving >> quintics, so assuming it is working correctly, this does indeed mean >> that these solutions cannot be represented by radicals. >> >> Aaron Meurer >> >> On Wed, Jun 5, 2013 at 9:02 AM, Chris Smith <[email protected]> wrote: >> >>>> [w.n(3) for w in solve(S('x**5 + x**3 + 1'))] >> > [-0.838, -0.218 - 1.17*I, -0.218 + 1.17*I, 0.637 - 0.665*I, 0.637 + >> 0.665*I] >> > >> > -- >> > 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 http://groups.google.com/group/sympy?hl=en-US. >> > For more options, visit https://groups.google.com/groups/opt_out. >> > >> > >> >> -- >> 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 http://groups.google.com/group/sympy?hl=en-US. >> For more options, visit https://groups.google.com/groups/opt_out. >> >> >> > -- > 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 http://groups.google.com/group/sympy?hl=en-US. > For more options, visit https://groups.google.com/groups/opt_out. > > > -- 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 http://groups.google.com/group/sympy?hl=en-US. For more options, visit https://groups.google.com/groups/opt_out.
