Examples of some \({\mathbb{Q}}\)-admissible groups
type: Item
instance of: scholarly article
title: Examples of some \({\mathbb{Q}}\)-admissible groups
zbMATH Open document ID: 0619.12007
DOI: 10.1016/0022-314X(87)90079-5
author: Walter Feit
author: Paul Vojta
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Recall that, if k is a number field, a finite group G is called k- admissible if G is isomorphic to Gal(K/k) for some number field K that is a maximal subfield of a finite-dimensional division algebra with center k. The authors prove the \({\mathbb{Q}}\)-admissibility of the groups \(A_ 6\), \(A_ 7\), \(\tilde A_ 7\), and thus complete the list of alternating groups which may be \({\mathbb{Q}}\)-admissible. The admissibility of the groups G in question is proved by exhibiting the fields K (for which \(G\cong Gal(K/ {\mathbb{Q}}))\) as the splitting fields of certain generalized Laguerre polynomials. The coefficients of these polynomials are constructed from expressions that are defined using certain rational points on the elliptic curves \(y^ 2=15x(x^ 2-4)\) and \(y^ 2=105x(x^ 2-4)\). The rational points, themselves, were found with the aid of a computer. One rather amusing fact pointed out by the authors is that the theory of heights shows that the numerator or denominator of the constant term of the polynomial for \(A_ 7\) has approximately \(1{1/2}\) billion digits.
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 20F29
zbMATH DE Number: 4004310
zbMATH Keywords: \({bbfQ}\)-admissible group
zbMATH Keywords: k-adequate
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: elliptic curves
zbMATH Keywords: heights
reviewed by: Sheldon Kamienny
MaRDI profile type: Publication
cites work: \(\tilde A_5\) and \(\tilde A_7\) are Galois groups over number fields
cites work: Q4156850
cites work: The admissibility of \(A_5\)
cites work: Q4168661
cites work: Q3309989
cites work: Subfields of division rings. I
cites work: The Witt invariant of the form \(\text{Tr}(x^ 2)\)
description: scientific article; zbMATH DE number 4004310