2 problems

Let $K$ be an $S_n$-field for $n\geq 3$ and signature $(r_1,r_2)$. Let $\mathcal S$ be a finite set of local conditions, and let $|\mathcal S|$ denote the product of their local de…

Fix $d+1\geq3$ and a signature $(r_1,r_2)$. Let $L(X)^{r_2}$ be the set of $S_{d+1}$-fields $K$ with $X/2<|D_K|<X$, and let $L(X;\mathcal S)^{r_2}$ be the subset satisfying finitel…