On Infinite Unramified Extensions

Let k be a number field. A natural question is: Does k admit an infinite unramified extension? The answer is no, if the root discriminant of k is less than Odlyzko’s bounds. The answer is yes, if k fails the test of Golod-Shafarevic for a prime number p. In that case, we know that there exists an infinite unramified p-extension L over k. But generally it is fairly difficult to determin whether k admits an infinite unramified extension. For this problem we introduce the following unramified extensions of k: i) k∞ is the maximal unramified Galois extension of k. ii) k(1) denotes the Hilbert field of k, i.e., the maximal unramified abelian extension of k; its Galois group over k is isomorphic to the class group of k via the Artin map. More generally, let k(i) be the Hilbert field of k(i−1), i ≥ 1, where k(0) = k. Write kH = ∪k(i); kH is the Hilbert tower of k. We say that the Hilbert tower is finite (or stops) if [kH/k] <∞, or infinite otherwise. iii) For a prime number p, kp will be the p-Hilbert tower of k, that is to say the maximal p-extension of k contained in kH . We have the following inclusions: