An example of an infinite tamely ramified Hilbert tower (Q1383577)

The author gave a refinement of the Golod-Shafarevich Theorem in an earlier article [Nagoya Math. J. 150, 1-11 (1998)] which is now used to show that the field \(k=\mathbb{Q} (\sqrt {-3^* 13^* 61})\) has an infinite 2-tower of extensions unramified except at primes dividing 13. It is also shown that this example is related to the constant of Martinet, \(a(0,1)\). This constant is defined by \(a(0,1)= \lim_n \inf a_n^{1/2n}\), where \(a_n= \min \{| d_n|: d_n\) is the discriminant of a totally imaginary number field, \(K\), with \([K: \mathbb{Q}]= 2n\}\). This example shows either \(a(0,1) <48.78\) or that \(k\) has finite Hilbert 2-tower and an infinite 2-Hilbert tower ramified at only one place. Since the best known result is \(a(0,1)\leq 93\), it would be interesting to know whether or not \(k\) has finite Hilbert 2-tower.