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: