2 problems

Let $P_1,\ldots,P_n\in\mathbb{Q}[t]$ be pairwise distinct irreducible monic polynomials. Let $k_i=\mathbb{Q}[t]/(P_i(t))$ be the corresponding number fields, and let $a_i\in k_i$ d…

Let $k$ be a number field and let $K/k$ be a cyclic extension of degree $3$. For a polynomial $f\in k[t]$ of degree $6$, let $\mathcal{S}_{f}$ denote the variety defined by the nor…