I had a question which required me to show that $$\prod\limits_{\substack{1 \le r \le p\\ \ 1 \le s \le q}} \big( x - (\sqrt[p]{2}\zeta_{p}^s + \sqrt[q]{3}\zeta_{q}^r) \big)$$ is the minimal polynomial of $\sqrt[3]{2} + \sqrt[3]{3}$ over $\Bbb Q$. While I know this is the minimal polynomial in $\Bbb Q(\sqrt[3]{2},\sqrt[3]{3}, \zeta_{pq})$, my concern is that I have no idea how to show the product is a polynomial in $\Bbb Q[x]$. Is there some way to look at the coefficients of the final product and see they are rational?
2026-04-07 15:33:24.1775576004
Is $\prod (x - (\sqrt[p]{2}\zeta_{p}^s + \sqrt[q]{3}\zeta_{q}^r) )$ a polynomial with rational coefficients, p, q prime?
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