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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.

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    $\begingroup$ I don't understand by "each $L_n$ is distinct and in fact a $\mathbb Z/p$ extension". It seems to me that you have not assumed anything about the relationship among the $L_n$, so there is no way to prove they are distinct. For instance you could redefine $L_{n-1} = L_n$ for some $n$ and it would still satisfy the inequality. $\endgroup$
    – Will Sawin
    Commented Jul 18, 2018 at 14:51
  • $\begingroup$ Thanks! I realised it was ill framed, I have made the required edit! $\endgroup$
    – debanjana
    Commented Jul 18, 2018 at 15:10
  • $\begingroup$ Do you want a lower bound for the discriminant all $L_n$ or an upper bound for the discriminant of at least one $L_n$? $\endgroup$
    – Will Sawin
    Commented Jul 18, 2018 at 15:56
  • $\begingroup$ a lower bound for sure $\endgroup$
    – debanjana
    Commented Jul 18, 2018 at 16:06
  • $\begingroup$ Your best bet is almost certainly to simply bound the size of the class group using Brauer-Siegel en.wikipedia.org/wiki/Brauer%E2%80%93Siegel_theorem. $\endgroup$
    – Will Sawin
    Commented Jul 18, 2018 at 16:17

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