# H2

There exist an absolute constant \(C>0\) and an integer \(\ell_0\ge2\) such that, for every integer \(\ell\ge\ell_0\), putting
\[
t=\left\lfloor\frac{(\ell-1)^2}{100}\right\rfloor,
\]
there exist a totally real cyclic cubic field \(F/\mathbb Q\), distinct rational primes \(q_1,\ldots,q_t\), and a tower
\[
F=F_0\subset F_1\subset F_2\subset\cdots
\]
satisfying all the following properties.

1. Every \(q_b\) is congruent to \(1\pmod4\) and splits completely in every \(F_j\), for every \(j\ge0\).

2. Every \(F_j/F\) is finite Galois and everywhere unramified, and its Galois group is a \(3\)-group. Every \(F_j\) is totally real.

3. Writing \(f_j=[F_j:\mathbb Q]\), one has
\[
f_j\longrightarrow\infty,
\qquad
\operatorname{rd}(F_j)=\operatorname{rd}(F),
\qquad
\log\operatorname{rd}(F)\le C\ell\log\ell.
\]

For each fixed \(\ell\), the primes \(q_1,\ldots,q_t\) are chosen once and are independent of \(j\).

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.