Some \(\widetilde A_ n\)-extensions obtained from generalized Laguerre polynomials. (English) Zbl 0829.12004

Hajir, Farshid

J. Number Theory 50, No. 2, 206-212 (1995).

Let \(L(x,\lambda)\) be the \(n\)-th degree generalized Laguerre polynomial. It is proved that for \(n\geq 14\) the Galois group of \(f(x)=L(x,-2-n)\) is \(A_n\) if \(n\equiv 1\pmod 4\), and \(S_n\) otherwise. Moreover it is proved that the splitting field of \(f(x)\) can be embedded in a field with absolute Galois group isomorphic to \(\widetilde A_n\), the double cover of \(A_n\), if and only if \(n\equiv 1\pmod 8\).

Reviewer: Nuria Vila (Barcelona)

MSC:

12F12	Inverse Galois theory
11R32	Galois theory
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

embedding; inverse Galois theory; generalized Laguerre polynomial; Galois group