An explicit integral polynomial whose splitting field has Galois group $W\left({\mathbf{E}}_8\right)$

Jouve, Florent; Kowalski, Emmanuel; Zywina, David

Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 761-782.

En utilisant le principe selon lequel le polynôme caractéristique de matrices obtenues comme éléments d’un groupe réductif $\mathbf{G}$ sur $\mathbf{Q}$ a typiquement un corps de décomposition dont le groupe de Galois est le groupe de Weyl de $\mathbf{G}$, nous construisons un polynôme unitaire explicite de degré $240$, à coefficients entiers, dont le corps de décomposition a pour groupe de Galois le groupe de Weyl du groupe exceptionnel de type ${\mathbf{E}}_8$.

Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group $\mathbf{G}$ over $\mathbf{Q}$ typically have splitting field with Galois group isomorphic to the Weyl group of $\mathbf{G}$, we construct an explicit monic integral polynomial of degree $240$ whose splitting field has Galois group the Weyl group of the exceptional group of type ${\mathbf{E}}_8$.

Mots clés : Inverse Galois problem, Weyl group, exceptional algebraic group, random walk on finite group, characteristic polynomial

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