[seqfan] Re: Difficult problem

[Max Alekseyev]

On Sun, Dec 27, 2009 at 4:34 PM, Artur <grafix at csl.pl> wrote:

> We have real plane in 3D Cartesian space
> x(1-a)+(a^2-a)y+(2-a^4)z=0
> where a is single one real root of quintic polynomial a^5-a-1=0
> Does exist on this plane rational points (different as od x=y=z=0) ?

There are no rational points other than x=y=z=0 on this plane.
Imagine that there is such a point. Then
x(1-a)+(a^2-a)y+(2-a^4)z=0
can be viewed as degree-4 polynomial whose root is a.
However, since a^5-a-1 is irreducible over the rationals, no
polynomial of smaller degree with the root a exists.

Max