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Using the principle that characteristic polynomials of matrices obtained fromelements of a reductive group over a number field typically have splittingfield with Galois group isomorphic to its Weyl group, we construct an explicitmonic integral polynomial of degree 240 whose splitting field has Galois groupthe Weyl group of the exceptional group of type E_8.

An explicit integral polynomial whose splitting field has Galois group W(E_8)