Consider $F/\mathbb{Q}$, a number field. Let $S$ be a finite set of primes of $F$ containing the Archimedean primes. Let $n$ be any natural number and $L_n$ be a finite extension of $F$ such that $\dim_{\mathbb{Z}/p}(Cl_{S}(L_n)/p)\geq n$. Can one give a bound on the discriminant of $L_n$ as a function of $n$?

By an iterative application of the Grunwald-Wang theorem (and Prop 10.10.3 of Neukirch-Schmidt-Wingberg) one can show that each $L_n$ is distinct and in fact a $\mathbb{Z}/p$ extension. I am looking for a bound on the discriminant though.

Consider $F/\mathbb{Q}$, a number field. Let $S$ be a finite set of primes of $F$ containing the Archimedean primes. Let $n$ be any natural number and $L_n$ be a finite extension of $F$ such that $\dim_{\mathbb{Z}/p}(Cl_{S}(L_n)/p)\geq n$. Can one give a bound on the discriminant of $L_n$ as a function of $n$?

By an iterative application of the Grunwald-Wang theorem (and Prop 10.10.3 of Neukirch-Schmidt-Wingberg) one can show that there exists a sequence $\{L_n\}$ such that each $L_n$ is distinct and in fact a $\mathbb{Z}/p$ extension. I am looking for a bound on the discriminant though.