On Jun 25, 10:00 pm, davidloeffler <dave.loeff...@gmail.com> wrote:
> On Jun 25, 6:28 pm, egb <ebaza...@yahoo.com> wrote:
>
> > Hello!
>
> > I have a polynomial P, let's say P = x^3 - 139656*x^2 -
> > 59208339456*x - 1467625047588864.
>
> > K.<a> = NumberField(P)
>
> > Clearly a.charpoly() gives me P. I want to know if there is a
> > way to express the other two roots of P, besides a, as a polynomial
> > with rational coefficients in a.
>
> P has no other roots in K, except a. Why do you think it should?
Indeed, you have just shown that in this case there are no more roots
in K. The other two roots would lie in a quadratic extension of K.
The rule is that for an irreducible cubic, the roots can be expressed
in terms of eacho ther (as polynomials) if and only if the
discriminant is a square; which in this case it is not. This can be
rephrased (and proved easily) in the language of Galois theory.
John Cremona
>
> David
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