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Let $p(x) = \sum_{i=0}^{n} c_i x^i$ with $c_i \in \mathbb{A}$ with $q_i(c_i) = 0$ and all $q_i \in \mathbb{Q}[x]$ being minimal polynomials of the coefficients.

Let $r$ be a zero of $p(x)$. Is there an algorithm to compute the minimal polynomial of $r$, $m(x) \in \mathbb{Q}[x]$?

In the case where $c_i \in \mathbb{Q}$ it's easy. But I can't see a way in general for algebraic coefficients even when their minimal polynomials are known.

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    $\begingroup$ This is the subject of elimination theory, where you want to eliminate the variables $c_i$ and just leave an equation in terms of $x$. See e.g. math.stanford.edu/~vakil/216blog/FOAGnov2210p176-180.pdf for a basic treatment in terms of Chevalley's theorem on constructible sets. $\endgroup$
    – Kevin Casto
    Commented Jul 13 at 15:06
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    $\begingroup$ What is $\mathbb A$? I've seen it used for affine space, and occasionally for the ring of adeles. I'm guessing that you mean it to be the field of algebraic numbers, but since it's not standard notation, you should say that. And a more common notation for an algebraic closure of $\mathbb Q$ is $\overline{\mathbb Q}$. $\endgroup$
    – Joe Silverman
    Commented Aug 18 at 11:41

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One approach would be as follows: Let $c_i^{(j)},\,j=1,\ldots,\deg q_i$ be the conjugates of $c_i$. Denote $p_{j_0,\ldots,j_n}(x)=\sum_{i=0}^nc_i^{(j_i)}x^i$ and form the product $P(x)=\prod_{j_1,\ldots,j_n}p_{j_1,\ldots,j_n}(x)\in\mathbb Q[x]$. The minimal polynomial of $r$ is one of the factors of $P$ in $\mathbb Q[x]$. Which factor it is may depend on $r$ and not just on $q_0,\ldots,q_n$.

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