Question on determining the minimal polynomial for an algebraic quotient - MathOverflow

Question on determining the minimal polynomial for an algebraic quotient

2010-05-13T15:22:52Z

I need to determine the minimal polynomial for a quotient in (1).

(1) B = C / A

C is known as a root of a 36th degree polynomial and A is known as a root of a 24th degree polynomial.

However I have not been able to succeed in recovering the coefficients nor the degree of the polynomial for B.

Any suggestions? I have tried to use GP-Pari's algdep(number,power) command, but so far with little success, even though I know the decimal value of B to 10,018 digits.

Thanks for your help.

Randall

P.S. This is a repost after a suggestion

After working with the resultant method, I was able to successfully recover a 144th degree polynomial whose highest power term has the expected square coefficient. This polynomial was one of 3 polynomials factored from a 864th degree polynomial originally obtained.

I guessed 72nd degree, but it would have taken too long using GP-Pari's algdep(number,144) to recover the polynomial.

Thanks for your suggestions, I now have a valuable tool to help me work with algebraic vectors in R3.

Answer by Kevin Buzzard for Question on determining the minimal polynomial for an algebraic quotient

2010-05-13T15:30:46Z

Here's a suggestion. Use polcompositum(FA,FC) (with FA the min poly of A, FC the min poly of C) to find a number field K=Q(alpha) containing roots of both your polynomials, and then use lindep() to find a relation between 1,alpha,alpha^2,...,alpha^{d-1} and B. That will probably be much more efficient, because somehow you are using the knowledge of FA and FC this way, rather than just using algdep, which is throwing it away completely.

Answer by Gerry Myerson for Question on determining the minimal polynomial for an algebraic quotient

2010-05-14T01:55:25Z

Let $F$ be the polynomial for $A$, let $G$ be the polynomial for $C$. Consider the resultant of $x^{24}F(y/x)$ and $G(y)$. This will be a polynomial whose roots are all the numbers of the form $\gamma/\alpha$, where $\gamma$ (resp., $\alpha$) runs through the roots of $G$ (resp., $F$). The resultant is the determinant of a $60\times60$ matrix.