On the Galois groups of the exponential Taylor polynomials (Q1119968)

Let \(f_ n(X)\) be the polynomial \(1+x+x^ 2/2!+\dots+x^ n/n!\) over \({\mathbb{Q}}\). Then [cf. \textit{I. Schur}, Sitzungsber. Akad. Wiss. Berlin 1930, 443--449 (1930; JFM 56.0110.02)] the Galois group of \(f_ n(X)\) is the alternating group \(A_ n\) if 4 divides \(n\) and it is equal to the symmetric group \(S_ n\) otherwise. This paper gives a new elegant proof of this fact making efficient use of \(p\)-adic Newton polygons to find out things about the degrees of the factors of \(f_ n(X)\) over the \(p\)-adic completions \({\mathbb Q}_ p\).