Hilbert towers of cubic cyclic extensions of \(\mathbb{Q}\) (Q679462)

Using a homological approach to class field towers due to \textit{R. Schoof} [J. Reine Angew. Math. 372, 209-220 (1986; Zbl 0589.12011)], the author gives a sharpening of a criterion for infinite \(p\)-towers over a cubic cyclic extension of the rationals. The main result is that for a prime \(p\), different from 2 or 3, and cyclic cubic \(k/ {\mathbb Q}\), if the \(p\)-rank of the class group of \(k\) is at least 4, then \(k\) admits an infinite \(p\)-class field tower. This improves the Golod-Shafarevich result applied to this case, where the rank is required to be at least 6.