I searched for the roots of the sixth legendre polynomial
expressed using real radicals (see below)
o.k. this is clearly not possible
due to a theorem about the "Casus irreducibilis" for cubic polynomials
which states that
if all three roots are real,
then there is at least one which can't be expressed using real radicals.
search Google for proofs.
Regards
Andre
On Friday 25 January 2008, andre maute wrote:
> What i did
>
> -------- snip ----------
> p : 231*x^6/16-315*x^4/16+105*x^2/16-5/16;
>
> z : solve(p=0,x);
>
> z[1];
>
> z1 : radcan(z[1]), algebraic;
> z2 : ratsimp(z[1]), algebraic;
> z3 : ratsimp(rectform(z[1]));
> -------- snip ----------
>
> z1 and z2 contain nevertheless %i.
> z3 introduces trigonometric functions
> which i dislike as much as the %i
>
> i want an expression only involving radicals,
> no %i and no other auxilliary functions
>
> i'm not aware if the trigonometric expressions
> have a representation only involving radicals
> no %i and no other auxilliary functions
>
> what i have in mind
> is something like the following for the fifth legendre polynomial
>
> (%i10) p5:x^5/8-35*x^3/4+15*x/8;
> (%o10) 63*x^5/8-35*x^3/4+15*x/8;
> (%i11) z5:solve(p5 = 0,x);
> (%o11) [x = -sqrt(2*sqrt(70)+35)/(3*sqrt(7)),
> x = sqrt(2*sqrt(70)+35)/(3*sqrt(7)),
> x = -sqrt(35-2*sqrt(70))/(3*sqrt(7)),
> x = sqrt(35-2*sqrt(70))/(3*sqrt(7)),x = 0]
>
>
> Andre
>
> On Friday 25 January 2008, Stavros Macrakis wrote:
> > Did you try radcan and/or ratsimp(zz),algebraic?
> >
> > On 1/25/08, andre maute <andre.maute at gmx.de> wrote:
> > > On Friday 25 January 2008, you wrote:
> > > > Have you tried solve? This polynomial is the functional composition
> > > > of lower-degree polys so should easily be handled by solve. No need
> > > > for numerical solutions and identify. Take a look at the doc for
> > > > polydecomp if you're interested.
> > >
> > > solve gives long lists of roots involving %i
> > > it's known that the roots of the legendre polynomials are real,
> > > so what i want to see is no %i.
> > >
> > > polydecomp is no help either.
> > >
> > > Andre
> > >
> > > > -s
> > > >
> > > > On 1/25/08, andre maute <andre.maute at gmx.de> wrote:
> > > > > On Thursday 24 January 2008, Andrej Vodopivec wrote:
> > > > > > On Jan 22, 2008 5:04 AM, Jordi Guti?rrez Hermoso
> > > > > > <jordigh at gmail.com>
> > > > >
> > > > > wrote:
> > > > > > > If my calculations are correct, the roots of x^6 + 3*x^5 +
> > > > > > > 6*x^4 + 3*x^3 + 9*x +9 should all be expressible by radicals.
> > > > > > > In fact, they are all polynomials in terms of 2^(1/3) and a
> > > > > > > cube root of unity.
> > > > > > >
> > > > > > > How can I make Maxima tell me what the roots actually are?
> > > > > >
> > > > > > I don't know how to compute the roots in radicals, but I have
> > > > > > some code which can sometimes guess how they look like from their
> > > > > > numerical values:
> > > > > >
> > > > > > (%i1) load(identify)$
> > > > > > (%i2) x^6 + 3*x^5 + 6*x^4 + 3*x^3 + 9*x + 9$
> > > > > > (%i3) allroots(%)$
> > > > > > (%i4) identify(%);
> > > > > > (%o4)
> > >
> > > [x=(sqrt(3)*%i)/2+(3*2^(1/3)-3/2)/3,x=(3*2^(1/3)-3/2)/3-(sqrt(3)*%i)/2,
> > >x=
> > >
> > > > >.2
> > > > >
> > > > > >250982321872863*%i+(-3/4^(1/3)-3/2)/3,x=(-3/4^(1/3)-3/2)/3-
> > >
> > > .2250982321872863*%i,x=1.95714903975616*%i+(-3/4^(1/3)-3/2)/3,x=(-3/4^(
> > >1/
> > >
> > > > >3)
> > > > >
> > > > > >-3/2)/3-1.95714903975616*%i] (%i5) subst(first(%), %o2), ratsimp;
> > > > > > <-- you need to check if we
> > > > > > guessed correctly
> > > > > > (%o5) 0
> > > > > >
> > > > > > After you have two roots written with radicals you should be able
> > > > > > to solve the polynomial of degree 4 to get others.
> > > > > >
> > > > > > identify works mostly like the identify from maple but I think is
> > > > > > less powerfull. If you would find it interesting let me know.
> > > > >
> > > > > Perhaps you could check
> > > > > what your identify function gives for the sixth legendre
> > > > > polynomial.
> > > > >
> > > > > 231*x^6/16-315*x^4/16+105*x^2/16-5/16
> > > > >
> > > > > I have not seen the roots written in radicals, yet.
> > > > >
> > > > > Andre
> > > > >
> > > > > _______________________________________________
> > > > > Maxima mailing list
> > > > > Maxima at math.utexas.edu
> > > > > http://www.math.utexas.edu/mailman/listinfo/maxima
>
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