On infinite unramified extensions.
type: Item
instance of: scholarly article
title: On infinite unramified extensions.
zbMATH Open document ID: 1037.11074
author: Christian Maire
published in: Pacific Journal of Mathematics
publication date: 2000-06-22T00:00:00Z
review text: 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 \(2^\infty\).
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R21
zbMATH DE Number: 1463745
zbMATH Keywords: finite Hilbert tower
zbMATH Keywords: infinite unramified extension
reviewed by: Charles J. Parry
MaRDI profile type: Publication
full work available at URL: pjm.2000.192.135
OpenAlex ID: W1981904634
DOI: 10.2140/PJM.2000.192.135
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description: scientific article; zbMATH DE number 1463745