Science and Technology news... :: View topic

Guest · Posted: Sat Nov 01, 2008 4:11 pm Post subject: The Solvable Sextic x^6+3x+3 = 0

The quintic x^5-5x+12 = 0 is solvable. So I looked for solvable
irreducible sextic trinomials x^6+ax+b = 0. Turns out there are simple
examples, such as the one I found above. In fact, a general
parametrization can be given (discussed in another post).

For the special case x^6+ax+a = 0, for |a| < 1000, we only have:

x^6+3x+3 = 0
x^6+125x+125 = 0
x^6+320x+320 = 0

where they factor over the extension Sqrt[-3], Sqrt[5], Sqrt[5],
respectively, and,

x^6-49x-49 = 0

which does _not_ factor over a sqrt extension. (Are these four the
only ones with integral "a"?) The last can still be solved though
since it is just a composition of the two eqns:

-21-7v+5v^2 + vx + x^2 = 0
-7-7v+v^3 = 0

Such trinomial sextics have a parametrization as well.

P.S. What are the transitive groups of these (as numbered by Magma)?
(I know there are 15, with 12 as solvable.)

Tito

Guest

Oh, yeah, almost forgot, the simple sextic,

x^6+3x+5 = 0

is solvable as well (though does _not_ factor over a square root
extension).

It is a composition of:

-1-v+v^2+vx+x^2 = 0
-3-3v+v^3 = 0

Tito


www.GetXFactor.com Leading Technology, Science, Agriculture News and information	Part of the Identityscape.com network... getxfactor.com jmoodmusic.com smartbusinesschoices.com mintdepot.com lowfaresalways.com evangelicalview.com shoppingpodder.com soproudlywehail.com webnews.ws currenthumor.com

Science and Technology news... Forum Index -> Math - Symbolic Forum
Page 1 of 1