Monthly Archives: May 2014

Newton polygons and Galois groups

Issai Schur

Issai Schur

A famous result of David Hilbert asserts that there exist irreducible polynomials of every degree n over {\mathbf Q} having the largest possible Galois group S_n.  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 n^{\rm th} Laguerre polynomial L_n(x) = \sum_{j=0}^n (-1)^j \binom{n}{j} \frac{x^j}{j!} is irreducible and has Galois group S_n over {\mathbf Q}.

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:

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