# H1

There exist an absolute constant \(C>0\) and an integer \(\ell_0\ge2\) such that, for every integer \(\ell\ge\ell_0\), there exists a totally real cyclic cubic field \(F/\mathbb Q\) with the following properties.

Let \(F^{\mathrm{ur},3}\) be the maximal everywhere-unramified pro-\(3\) extension of \(F\), and put \(G=\operatorname{Gal}(F^{\mathrm{ur},3}/F)\). Then
\[
d(G)\ge\ell-1,
\qquad
\log\operatorname{rd}(F)\le C\ell\log\ell.
\]

Here \(d(G)\) is the minimum number of topological generators of \(G\).

Here \(\operatorname{rd}(L)=|\operatorname{Disc}(L)|^{1/[L:\mathbb Q]}\) is the root discriminant of a number field \(L\), and all logarithms are natural.