Question on determining the minimal polynomial for an algebraic quotient - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:32:19Z http://mathoverflow.net/feeds/question/24513 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24513/question-on-determining-the-minimal-polynomial-for-an-algebraic-quotient Question on determining the minimal polynomial for an algebraic quotient Randall 2010-05-13T15:22:52Z 2010-05-14T19:06:02Z <p>I need to determine the minimal polynomial for a quotient in (1).</p> <p>(1) B = C / A</p> <p>C is known as a root of a 36th degree polynomial and A is known as a root of a 24th degree polynomial.</p> <p>However I have not been able to succeed in recovering the coefficients nor the degree of the polynomial for B.</p> <p>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.</p> <p>Thanks for your help.</p> <p>Randall</p> <p>P.S. This is a repost after a suggestion</p> <p>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.</p> <p>I guessed 72nd degree, but it would have taken too long using GP-Pari's algdep(number,144) to recover the polynomial.</p> <p>Thanks for your suggestions, I now have a valuable tool to help me work with algebraic vectors in R3.</p> http://mathoverflow.net/questions/24513/question-on-determining-the-minimal-polynomial-for-an-algebraic-quotient/24514#24514 Answer by Kevin Buzzard for Question on determining the minimal polynomial for an algebraic quotient Kevin Buzzard 2010-05-13T15:30:46Z 2010-05-13T15:30:46Z <p>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.</p> http://mathoverflow.net/questions/24513/question-on-determining-the-minimal-polynomial-for-an-algebraic-quotient/24568#24568 Answer by Gerry Myerson for Question on determining the minimal polynomial for an algebraic quotient Gerry Myerson 2010-05-14T01:55:25Z 2010-05-14T01:55:25Z <p>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. </p>