# Irreducibility over the field of fractions of a quotient of a polynomial ring

Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell - 1$ and $z_0, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, non-homogeneous) polynomials

$q_i = z_{\ell}x_i^r - z_ix_{\ell}^r - (z_{\ell} - z_i)$ for $0 \leq i < \ell$ and

$p = z_\ell x_0^{r + k} - x_1\cdots x_k(1 - x_{\ell}^r)$

in $R = \mathbb{C}[x_1, \ldots, x_{\ell}]$. Define $I$ to be the ideal

$I = (q_1, \ldots, q_{\ell - 1}) \subset R$ and let $K$ be the field of fractions of $R/I$. Then $p$ and $q_0$ can be regarded as polynomials in $(R/I)[x_0] \cong \mathbb{C}[x_0, \ldots, x_{\ell}]/(q_1, \ldots, q_{\ell - 1})$, thus we can also regard them as polynomials in $K[x_0]$.

I have an argument that proves the irreducibility of $p$ and $q_0$ over $R/I$ and after having computed examples using Sage/Singular, I think that $p$ and $q_0$ are both irreducible over $K$ as well, but I don't see how to prove this. Proving it directly by writing it as a product in $K[x_0]$ and trying to work out what the factors could be didn't work and as $R/I$ is not a GCD domain, one can't use Gauss' lemma. Does anyone know some argument that could be applied here to deduce the irreducibility of $p$ and $q_0$ over $K$?

Note: As this is a problem that arose when trying to use the Gianni-Trager-Zacharias primality test, we can assume (by induction) that $I$ is a prime ideal in $R$. See here for the original problem.