Given two algebraic conjugates $\alpha,\beta$ and their minimal polynomial, find a polynomial that vanishes at $\alpha\beta$ in a efficient way

Inspired by this question, I was wondering about the following problem.

$\alpha,\beta,\gamma,\ldots$ are the roots of an irreducible
polynomial over $\mathbb{Q}$. How to compute the coefficients of $$
(x-\alpha\beta)(x-\alpha\gamma)\cdot \ldots\cdot (x-\beta\gamma)\cdot\ldots, $$ i.e. the candidate minimal polynomial
of $\alpha\beta$, in the most efficient way?

My approach is based on a simple lemma and a general fact. If $p_m$ is the power sum $\alpha^m+\beta^m+\gamma^m+\ldots$ and $e_i$ is the $i$-th elementary symmetric polynomial, the identity
$$ \exp\left(-\sum_{m\geq 1}\frac{p_m}{m}x^m\right) = \sum_{r\geq 0}(-1)^r e_r\,x^r$$
gives a way to compute $p_1,p_2,p_3,\ldots$ from $e_1,e_2,e_3,\ldots$ through the Taylor series of a logarithm ($\text{LOG}$) as well as $e_1,e_2,e_3,\ldots$ from $p_1,p_2,p_3,\ldots$ through the Taylor series of an exponential ($\text{EXP}$). Moreover, the characteristic polynomial of the sequence $\{p_k\}_{k\geq 0}$ is exactly the minimal polynomial of $\alpha$, hence we may compute $p_{n+1},p_{n+2},\ldots,p_{n+m}$ from $p_1,p_2,\ldots,p_n$ with a simple recursive approach (I will call this procedure $\text{CH}$, from Cayley-Hamilton). $\text{L}$ will be my previously mentioned lemma, i.e.
$$ p_k(\alpha\beta,\alpha\gamma,\ldots) = \frac{1}{2}\left(p_k^2-p_{2k}\right).$$
Now my algorithm goes as follows:

(1). $$ e_i(\alpha,\beta,\gamma,\ldots)_{i\leq n}\quad\xrightarrow{\text{LOG}}\quad p_i(\alpha,\beta,\gamma,\ldots)_{i\leq n}$$
(2). $$ p_i(\alpha,\beta,\gamma,\ldots)_{i\leq n}\quad\xrightarrow{\text{CH}}\quad p_i(\alpha,\beta,\gamma,\ldots)_{i\leq n(n-1)}$$
(3). $$ p_i(\alpha,\beta,\gamma,\ldots)_{i\leq n(n-1)}\quad\xrightarrow{\text{L}}\quad p_i(\alpha\beta,\alpha\gamma,\ldots)_{i\leq \binom{n}{2}}$$
(4). $$ p_i(\alpha\beta,\alpha\gamma,\ldots)_{i\leq \binom{n}{2}}\quad\xrightarrow{\text{EXP}}\quad e_i(\alpha\beta,\alpha\gamma,\ldots)_{i\leq \binom{n}{2}}.$$

Solutions Collecting From Web of "Given two algebraic conjugates $\alpha,\beta$ and their minimal polynomial, find a polynomial that vanishes at $\alpha\beta$ in a efficient way"