Some Ãn-Extensions Obtained from Generalized Laguerre Polynomials Original Research Article

For an integer n ≥ 14, we prove: (1) The Galois group of the polynomial ƒ(x) = ∑nj = 0 ((n − j + 1)/j!) xj is An if n ≡ 1 (mod 4), and Sn otherwise. (2) The splitting field of ƒ(x) can be embedded in a field with absolute Galois group isomorphic to Ãn, the double cover of An, if and only if n ≡ 1 (mod 8).