Some explicit integral polynomials with Galois group \(W(E_8)\)
type: Item
instance of: scholarly article
title: Some explicit integral polynomials with Galois group \(W(E_8)\)
zbMATH Open document ID: 1266.12003
author: Tetsuji Shioda
published in: Proceedings of the Japan Academy. Series A
publication date: 2009-12-22T00:00:00Z
review text: This paper can be seen as a natural complement to the author's paper ``Mordell-Weil lattices: theory and applications. A point of contact of algebra, geometry, and computer'' [Sugaku Expo. 7, No. 1, 19--41 (1994); translation from Sūgaku 43, No. 2, 97--114 (1991; Zbl 0746.14008)] and simultaneously as an alternative solution to the problem solved in [the reviewer, \textit{E. Kowalski} and \textit{D. Zywina}, ``An explicit integral polynomial whose splitting field has Galois group \(W(E_8)\)'', J. Théor. Nombres Bordx. 20, No. 3, 761--782 (2008; Zbl 1200.12003)] and [\textit{A. Várilly-Alvarado} and \textit{D. Zywina}, ``Arithmetic \(E_8\) lattices with maximal Galois action'', LMS J. Comput. Math. 12, 144--165 (2009; Zbl 1252.11055)]. The problem raised consists in the explicit construction of a Galois extension of the rationals with Galois group the Weyl group \(W(E_8)\) of the root lattice \(E_8\). This problem is solved theoretically by the author in his paper cited above. The idea is to first consider the cubic curve \[ E_\lambda: y^2=x^3+\left(\sum_{i=0}^3p_it^i\right)x+\left(\sum_{j=0}^3q_jt^j\right)+t^5\,, \] defined over \({\mathbb Q}[\lambda][t]\), where \(\lambda=(p_0,\dots,p_3,q_0,\dots,q_3)\). If \(k\) denotes a fixed algebraic closure of \({\mathbb Q}(\lambda)\) then it is known that we have a surjective Galois representation \[ \rho: \text{Gal}(k/{\mathbb Q}(\lambda))\rightarrow \Aut(E(k(t))\simeq W(E_8)\,. \] This argument combined with Hilbert's irreducibility theorem provides one with infinitely many \(W(E_8)\)-extensions of \({\mathbb Q}\) (via the specialization \(\lambda\rightarrow \lambda_0\in\mathbb Q^8\)). In the paper under review the author shows that \(\lambda\rightarrow (1,\dots, 1)\) (or \(\lambda\rightarrow (2,1\dots, 1)\)) is a suitable specialization for this purpose. Let \(E/\mathbb Q(t)\) be the elliptic curve obtained via the specialization \(\lambda\rightarrow (1,\dots, 1)\). There are exactly \(240\) \(\overline{\mathbb Q}(t)\)-ra…
reviewed by: Florent Jouve
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 5651158
zbMATH Keywords: explicit inverse Galois problem
zbMATH Keywords: Weyl group
zbMATH Keywords: Mordell-Weil lattices
MaRDI profile type: Publication
full work available at URL: pjaa.85.118
OpenAlex ID: W2047469100
cites work: Q3684278
cites work: Q4039784
cites work: An explicit integral polynomial whose splitting field has Galois group \(W(E_8)\)
cites work: Q3992577
cites work: Q5203144
cites work: Q4011133
cites work: Construction of elliptic curves with high rank via the invariants of the Weyl groups
cites work: Gröbner basis, Mordell-Weil lattices and deformation of singularities. I
cites work: Arithmetic E8 Lattices with Maximal Galois Action
DOI: 10.3792/PJAA.85.118
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description: scientific article; zbMATH DE number 5651158