Some explicit integral polynomials with Galois group \(W(E_8)\)
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 \(240\) \(\overline{\mathbb Q}(t)\)-rational points on \(E\) of a particular form involving a parameter \(v\) that satisfies \(\Psi(v)=0\) for some integral polynomial \(\Psi(x)=F(x^2)\) of degree \(240\). To prove that the Galois group of the splitting field of \(\Psi\) over \({\mathbb Q}\) is isomorphic to \(W(E_8)\) the author chooses suitable prime numbers \(p_1, p_2\) such that the factorisation type of \(\Psi \pmod {p_1}\) and \(\Psi \pmod {p_2}\) gives rise to two conjugacy classes of \(W(E_8)\) that are not simultaneously contained in any maximal subgroup of \(W(E_8)\). This argument was already used in [the reviewer et al., op. cit.]. It turns out one can choose \(p_1=5\), \(p_2=17\). The paper contains the (reasonably long) list of the coefficients of the polynomial \(F\).
scientific article; zbMATH DE number 5651158
scientific article