On infinite unramified extensions.
This article is concerned with fields that have a finite Hilbert tower, but an infinite unramified extension. For example, if \(\ell= 17380678572159893\), \(q_1= 16747\) and \(k= \mathbb{Q} (\sqrt {lq_1q_2})\) with \(q_2= 1051\) or one of 7 other primes then the Hilbert tower stops at \(\mathbb{Q} (\sqrt {\ell}, \sqrt {q_1 q_2})\) since this field has class number 1. However, if \(P(x)= x^7- 3x^6- 13x^5+ 28x^4+ 42x^3- 47x^2- 31x+ 12\) and \(K= \mathbb{Q}(\theta)\) where \(\theta\) is a root of \(P(x)\) and \(M= Kk\) then \(M\) has an infinite 2-Hilbert tower and \(M/k\) is unramified so \(k\) has an infinite unramified extension. The basic idea is to use the Golod-Shafarevich criterion [\textit{J. Martinet}, Invent. Math. 44, 65--73 (1978; Zbl 0369.12007)] to construct a field \(K\) which has an infinite \(p\)-Hilbert tower. Then a result from Kummer theory is used which requires the ramification indices in \(K/\mathbb{Q}\) to divide those in \(k/\mathbb{Q}\), to show that \(Kk/k\) is unramified. Hence \(k\) has an infinite unramified extension. Similar examples to the one given above are described generated by polynomials of degrees 17, 11 and 9. Finally, the Chebotarev Density Theorem is used to prove that there exist infinitely many quadratic fields (both real and imaginary) with a finite 2-Hilbert tower, but an infinite unramified extension of degree \(2^\infty\).
scientific article; zbMATH DE number 1463745