An algebraic number is defined as a root of a polynomial with rational coefficients.
It is known that every algebraic number $\alpha$ has a unique minimal polynomial, the monic polynomial with rational coefficients of smallest degree of which $\alpha$ is a root.
It is also known that the algebraic numbers are algebraically closed, meaning that any polynomial with algebraic numbers as coefficients has algebraic roots.
My question is:
Given the root of a degree $d$ uni-variate polynomial with $n$ terms with algebraic coefficients, is there an efficient (i.e. polynomial time in $d$ and $n$) algorithm to generate its minimal polynomial?
I have searched the literature, but found very little work on algebraic number theory concerned with computational complexity. The closest result I found is:
Theorem 1.4.7 in Lovasz's book:
but I don't think this (quite) answers my question.