Prev: Boundness of solutions of ODE systems
Next: formalizing what math assumed as the definition of "finite-number"#424 Correcting Math
From: I.M. Soloveichik on 18 Feb 2010 05:34
> Explicit example:
> An irreducible monic equation with integer
> coefficients
> of degree n, Galois group with 2n elements but not
> dihedral and at least 4 real roots (and both real and
> complex roots). This means n > 6.


x^8-9*x^4+9 has 4 real roots and galois of order 16
generated by
(1 5)(3 7), (1 4 5 8)(2 3 6 7), (2 4 6 8)(1 3 5 7)
From: I.M. Soloveichik on 18 Feb 2010 05:47
x^6+3*x^4-1
has 2 real roots and galois group generated by
(1 4)(2 5), (2 4 6)(1 3 5)
From: I.M. Soloveichik on 18 Feb 2010 06:33
> Do you have an example with n is odd?

x^9+3*x^6-3*x^3+1 has 1 real root and galois group generated by
(1 2 9)(3 4 5)(6 7 8),(1 2)(3 6)(4 8)(5 7),(3 6 9)(1 4 7)(2 5 8)
of order 18.
From: secondmouse on 19 Feb 2010 05:37
On Feb 18, 9:33 pm, "I.M. Soloveichik" <imsol...(a)yahoo.com> wrote:
> > Do you have an example with n is odd?
>
> x^9+3*x^6-3*x^3+1 has 1 real root and galois group generated by
> (1 2 9)(3 4 5)(6 7 8),(1 2)(3 6)(4 8)(5 7),(3 6 9)(1 4 7)(2 5 8)
> of order 18.

But the OP wanted at least 4 reals.
This can't occur when n=9.
> Explicit example:
> An irreducible monic equation with integer coefficients
> of degree n, Galois group with 2n elements but not dihedral and at least 4 real roots (and both real and complex roots). This means n > 6.
From: Derek Holt on 19 Feb 2010 08:52
On 19 Feb, 10:37, secondmouse <paul.timm...(a)btinternet.com> wrote:
> On Feb 18, 9:33 pm, "I.M. Soloveichik" <imsol...(a)yahoo.com> wrote:
>
> > > Do you have an example with n is odd?
>
> > x^9+3*x^6-3*x^3+1 has 1 real root and galois group generated by
> > (1 2 9)(3 4 5)(6 7 8),(1 2)(3 6)(4 8)(5 7),(3 6 9)(1 4 7)(2 5 8)
> > of order 18.
>
> But the OP wanted at least 4 reals.
> This can't occur when n=9.

The OPs post consisted of the single question:

> Do you have an example with n is odd?

He has asked for so many examples with so many different properties
that I am surprised you can be so certain!

Derek Holt.
First  |  Prev  |  Next  |  Last
Pages: 1 2 3 4 5 6 7 8 9 10
Prev: Boundness of solutions of ODE systems
Next: formalizing what math assumed as the definition of "finite-number"#424 Correcting Math