A famous result of David Hilbert asserts that there exist irreducible polynomials of every degree over having the largest possible Galois group . However, Hilbert’s proof, based on his famous irreducibility theorem, is non-constructive. Issai Schur proved a constructive (and explicit) version of this result: the Laguerre polynomial is irreducible and has Galois group over .

In this post, we give a simple proof of Schur’s result using the theory of Newton polygons. The ideas behind this proof are due to Robert Coleman and are taken from his elegant paper *On the Galois Groups of the Exponential Taylor Polynomials*. (Thanks to Farshid Hajir for pointing out to me that Coleman’s method applies equally well to the Laguerre polynomials.) Before we begin, here is a quote from Ken Ribet taken from the comments section of this post: