An explicit integral polynomial whose splitting field has Galois group   W(E_8)
  Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group over a number field typically have splitting field with Galois group isomorphic to its Weyl group, 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 E_8. 
F. Jouve, E. Kowalski, D. Zywina