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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
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