Galois number fields with small root discriminant
type: Item
instance of: scholarly article
title: Galois number fields with small root discriminant
zbMATH Open document ID: 1163.11079
author: John W. Jones
author: David P. Roberts
published in: Journal of Number Theory
publication date: 2009-03-09T00:00:00Z
review text: Let \(K\) be a Galois extension of \({\mathbb Q}\) with (finite) group \(G\). For \(x\in K\), one has \(\sum_{g\in G} g(x)\in {\mathbb Q}\), and so there exists a map \(\text{tr}: K\rightarrow {\mathbb Q}\) defined by the rule \(x\mapsto\sum_{g\in G} g(x)\). This is the trace map of \(K/{\mathbb Q}\). There is a non-degenerate symmetric bilinear form \(B: K\times K\rightarrow {\mathbb Q}\) defined as \(B(x,y)=\text{tr}(xy)\) for \(x,y\in K\). Let \(R\) be the integral closure of \({\mathbb Z}\) in \(K\). Then \(B: R\times R\rightarrow {\mathbb Z}\). Since \(R\) is a finitely generated torsion-free module over the PID \({\mathbb Z}\), \(R\) is \({\mathbb Z}\)-free of rank \(N=| G|\). Let \(\{x_1,x_2,\dots,x_N\}\) denote a \({\mathbb Z}\)-basis for \(R\). Let \(A\) be the \(N\times N\) matrix whose \(i,j\)th entry is \(B(x_i,x_j)=\text{tr}(x_ix_j)\), \(1\leq i,j\leq N\). Then the discriminant of \(K\), \(D(K)\), is the ideal of \({\mathbb Z}\) generated by the non-zero integer \(\text{det}(A)\). We identify \(D(K)\) with the integer \(\text{det}(A)\). Certainly, \(D({\mathbb Q})=1\), and by Minkowski's Theorem, \(| D(K)| > 1\) whenever \(K\not = {\mathbb Q}\). The root discriminant of \(K\) is defined as \(d(K)=| D(K)|^{1\over N}\). Let \(C\geq 1\) be a real number, and let \(G\) be a finite group. Let \({\mathcal K}(G,C)\) denote the collection of all Galois extensions with group isomorphic to \(G\) for which \(d(K)\leq C\). For example, let \(n\geq 1\) be an integer, and let \(\zeta_n\) denote a primitive \(n\)th root of unity. For a rational prime \(p\), \(d({\mathbb Q}(\zeta_p))=p^{p-2\over p-1}<p\), and so, \({\mathbb Q}(\zeta_p)\in {\mathcal K}(C_{p-1},p)\) where \(C_{p-1}\) denotes the cyclic group of order \(p-1\). A natural question arises: Can one compute all of the field extensions in \({\mathcal K}(C_{p-1},p)\)? This is a reasonable question since for a fixed group \(G\) and fixed constant \(C\geq 1\), it is known that \({\mathcal K}(G,C)\) is finite. In t…
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11Y70
zbMATH DE Number: 5526764
zbMATH Keywords: Galois number fields
zbMATH Keywords: root discriminant
zbMATH Keywords: Serre constant
reviewed by: Robert G. Underwood
describes a project that uses: PARI/GP
describes a project that uses: ecdata
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.05.001
OpenAlex ID: W2084967088
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DOI: 10.1016/J.JNT.2006.05.001
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description: scientific article; zbMATH DE number 5526764