find the relation $\beta = f(\alpha)$ when $\beta \in Q(\alpha)$

Question

I am interested in finding the relation $\beta = f(\alpha)$ when $\beta \in Q(\alpha)$ and their minimal polynominals are given.

For example, let $\alpha$ be $exp(2\pi i/7)$ and $\beta$ be a root of $F(y)=y^6-7y^5+49y^4-441y^3+1715y^2+2401y+2401$.

There are some rational integers $a,b,c,d,e,f$ such that $F(a\alpha^5+b\alpha^4+c\alpha^3+d\alpha^2+e\alpha+f)=0$.
Are there some procedures to find such $a,b,c,d,e,f$?

Answer 1

Practically, I learned to use command in PARI/GP such as

nffactor(t^6+t^5+t^4+t^3+t^2+t+1, y^6-7y^5+49y^4-441y^3+1715y^2+2401*y+2401)

to factor $F(y)$ in the number field $\mathbb{Q}[t]/(t^6+t^5+t^4+t^3+t^2+t+1)$. I can obtain the relation from its $1$-degree factor $(y-f(t))$.