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SUMMARY:Hugo Chapdelaine (Université Laval)
DTSTART:20221031T180000Z
DTEND:20221031T190000Z
DTSTAMP:20260423T021053Z
UID:NTC/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTC/2/">Comp
 utation of Galois groups via permutation group theory</a>\nby Hugo Chapdel
 aine (Université Laval) as part of Lethbridge number theory and combinato
 rics seminar\n\n\nAbstract\nIn this talk we will present a method to study
  the Galois group of certain polynomials defined over $\\Q$.\nOur approach
  is similar in spirit to some previous work of F. Hajir\, who studied\, mo
 re than a decade ago\, the generalized Laguerre polynomials using a simila
 r approach.\nFor example this method seems to be well suited to study the 
 Galois groups of Jacobi polynomials (a classical family of orthogonal poly
 nomials with two parameters --- three if we include the degree). Given a p
 olynomial $f(x)$ 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 som
 etimes succeed in showing that the Galois group of $f$ is not solvable 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 hav
 e 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 tha
 t\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 distinct and 
 small primes. We hope that such effective prime existence results are with
 in the reach of the current techniques used in analytic number theory.\n
LOCATION:https://researchseminars.org/talk/NTC/2/
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