Re: Galois resolvent

*To*: mathgroup at smc.vnet.net*Subject*: [mg96902] Re: [mg96892] Galois resolvent*From*: "Kent Holing" <KHO at StatoilHydro.com>*Date*: Fri, 27 Feb 2009 06:09:54 -0500 (EST)*References*: <200902261302.IAA26707@smc.vnet.net> <op.upy30iqgtgfoz2@bobbys-imac.local>

R(t) = t^3 + 2B t^2 + (B^2 - 4D)t - C^2 = 0. -----Original Message----- From: DrMajorBob [mailto:btreat1 at austin.rr.com] Sent: 26. februar 2009 20:21 To: Kent Holing; mathgroup at smc.vnet.net Subject: [mg96902] Re: [mg96892] Galois resolvent The link includes illegible items such as the question marks in R(t) = t^3 + 2B t^2 + (B^2 ? 4D)t ? C^2 = 0. Hence, I have no idea what you're asking. Bobby On Thu, 26 Feb 2009 07:02:29 -0600, Kent Holing <KHO at statoil.com> wrote: > Forthe quartic (*) x^4 + Bx^2 + Cx + D = 0 for integers B, C and D, > assume that as for the case C = 0 that all its roots are classically = > contructible also for the case C /= 0. > > We can then show that the equation (*) is cyclic (i.e. its Galois > group = Z4) iff the splitting field of its Descartes resolvent is E = = > Q[Sqrt[t0] /= Q for t0 the one and only integer roots t0 of the > resolvent. For details, see > http://mathforum.org/kb/thread.jspa?threadID=1903146. > > If the quartic (*) is cyclic, it should be possible using the above to > explicitly construct the so-called Galois resolvents of (*): The roots > x1, x2, x3 and x4 of the quartic (*) can be given by polynomials of r > with degree less or equal to 3 with rational coefficients for r an > arbitrarily root of the quartic. (I.e. the splitting field of the > quartic (*) when cyclic is Q[r] for r a root.) > > Can somebody, using Mathematica, explicitly determine these polynomial > representations of the roots of the quartic (*). The case C = 0 is easy. > But the case C /= 0 is indeed messy. > > Kent Holing, > NORWAY > -- DrMajorBob at bigfoot.com ------------------------------------------------------------------- The information contained in this message may be CONFIDENTIAL and is intended for the addressee only. Any unauthorised use, dissemination of = the information or copying of this message is prohibited. If you are not the addressee, please notify the sender immediately by return e-mail and = delete this message. Thank you.

**References**:**Galois resolvent***From:*Kent Holing <KHO@statoil.com>