Computation of Galois groups via permutation group theory

Abstract: In this talk we will present a method to study the Galois group of certain polynomials defined over $\Q$. Our approach is similar in spirit to some previous work of F. Hajir, who studied, more than a decade ago, the generalized Laguerre polynomials using a similar approach. For example this method seems to be well suited to study the Galois groups of Jacobi polynomials (a classical family of orthogonal polynomials with two parameters --- three if we include the degree). Given a polynomial $f(x)$ with rational coefficients of degree $N$ over $\Q$, the idea consists in finding a good prime $p$ and look at the Newton polygon of $f$ at $p$. Then combining the Galois theory of local field over $\Q_p$ and some classical results of the theory of permutation of groups we sometimes succeed in showing that the Galois group of $f$ is not solvable or even isomorphic to $A_N$ or $S_N$ ($N\geq 5$).

The existence of a good prime $p$ is subtle. In order to get useful results one would need to have some "effective prime existence results". As an illustration, we would like to have an explicit constant $C$ (not too big) such that for any $N>C$, there exists a prime $p$ in the range $N < p < \frac{3N}{2}$ such that gcd$(p-1,N)= 1 \text{ or } 2$ (depending on the parity of $N$). Such a result is not so easy to get when $N$ is divisible by many distinct and small primes. We hope that such effective prime existence results are within the reach of the current techniques used in analytic number theory.

combinatoricsnumber theory

Audience: researchers in the topic

Lethbridge number theory and combinatorics seminar