Re: Galois resolvent
- To: mathgroup at smc.vnet.net
- Subject: [mg96909] Re: [mg96892] Galois resolvent
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Fri, 27 Feb 2009 06:11:16 -0500 (EST)
- References: <200902261302.IAA26707@smc.vnet.net>
- Reply-to: drmajorbob at bigfoot.com
So you need these roots? Solve[t^3 + 2 B t^2 + (B^2 - 4 D) t - C^2 == 0, t] {{t -> -((2 B)/3) - (2^(1/3) (-B^2 - 12 D))/( 3 (2 B^3 + 27 C^2 - 72 B D + Sqrt[ 4 (-B^2 - 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3)) + (2 B^3 + 27 C^2 - 72 B D + Sqrt[ 4 (-B^2 - 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^(1/3)/( 3 2^(1/3))}, {t -> -((2 B)/3) + ((1 + I Sqrt[3]) (-B^2 - 12 D))/( 3 2^(2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[ 4 (-B^2 - 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3)) - ((1 - I Sqrt[3]) (2 B^3 + 27 C^2 - 72 B D + Sqrt[ 4 (-B^2 - 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^(1/3))/( 6 2^(1/3))}, {t -> -((2 B)/3) + ((1 - I Sqrt[3]) (-B^2 - 12 D))/( 3 2^(2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[ 4 (-B^2 - 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3)) - ((1 + I Sqrt[3]) (2 B^3 + 27 C^2 - 72 B D + Sqrt[ 4 (-B^2 - 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^(1/3))/( 6 2^(1/3))}} or these? Solve[x^4 + B x^2 + C x + D == 0, x] // Simplify {{x -> (1/( 2 Sqrt[6]))(\[Sqrt](-4 B + ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) + 2^( 2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3)) - \[Sqrt](-8 B - ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) - 2^( 2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) - (12 Sqrt[6] C)/(\[Sqrt](-4 B + ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^(1/3) + 2^(2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^(1/3)))))}, {x -> (1/( 2 Sqrt[6]))(\[Sqrt](-4 B + ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) + 2^(2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3)) + \[Sqrt](-8 B - ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) - 2^( 2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) - (12 Sqrt[6] C)/(\[Sqrt](-4 B + ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^(1/3) + 2^(2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^(1/3)))))}, {x -> -(1/( 2 Sqrt[6]))(\[Sqrt](-4 B + ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) + 2^( 2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3)) + \[Sqrt](-8 B - ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) - 2^( 2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) + (12 Sqrt[6] C)/(\[Sqrt](-4 B + ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^(1/3) + 2^(2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^(1/3)))))}, {x -> (1/( 2 Sqrt[6]))(-\[Sqrt](-4 B + ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) + 2^( 2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3)) + \[Sqrt](-8 B - ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) - 2^( 2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^( 1/3) + (12 Sqrt[6] C)/(\[Sqrt](-4 B + ( 2 2^(1/3) (B^2 + 12 D))/(2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^(1/3) + 2^(2/3) (2 B^3 + 27 C^2 - 72 B D + Sqrt[-4 (B^2 + 12 D)^3 + (2 B^3 + 27 C^2 - 72 B D)^2])^(1/3)))))}} Bobby On Fri, 27 Feb 2009 01:40:41 -0600, Kent Holing <KHO at statoilhydro.com> wrote: > 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: 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. -- DrMajorBob at bigfoot.com
- References:
- Galois resolvent
- From: Kent Holing <KHO@statoil.com>
- Galois resolvent