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SUMMARY:Hugo Chapdelaine (Université Laval)
DTSTART:20221031T180000Z
DTEND:20221031T190000Z
DTSTAMP:20260404T172830Z
UID:NTC/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTC/2
 /">Computation of Galois groups via permutation group theory</a>\nby Hugo 
 Chapdelaine (Université Laval) as part of Lethbridge number theory and co
 mbinatorics seminar\n\n\nAbstract\nIn this talk we will present a method t
 o study the Galois group of certain polynomials defined over $\\Q$.\nOur a
 pproach is similar in spirit to some previous work of F. Hajir\, who studi
 ed\, more than a decade ago\, the generalized Laguerre polynomials using a
  similar approach.\nFor example this method seems to be well suited to stu
 dy the Galois groups of Jacobi polynomials (a classical family of orthogon
 al polynomials with two parameters --- three if we include the degree). Gi
 ven 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 p
 olygon 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 solva
 ble or even isomorphic to $A_N$ or $S_N$ ($N\\geq 5$).\n\nThe 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 a
 ny $N>C$\, there exists a prime $p$ in the range $N < p < \\frac{3N}{2}$ s
 uch that\ngcd$(p-1\,N)= 1 \\text{ or } 2$ (depending on the parity of $N$)
 . Such a result is not so easy to get when $N$ is divisible by many distin
 ct and small primes. We hope that such effective prime existence results a
 re within the reach of the current techniques used in analytic number theo
 ry.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/2/
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