Some generalized Laguerre polynomials whose Galois groups are the alternating groups. (English) Zbl 0693.12009

Gow, R.

J. Number Theory 31, No. 2, 201-207 (1989).

The author considers one kind of generalized Laguerre polynomials
\[ F_n = (-1)^n n!\sum^n_{m=0}\binom{2n}{n-m}\frac{(-X)^m}{m!}\in\mathbb Z[X] \]
and shows that their Galois groups over \(\mathbb Q\) are the alternating group \(A_n\) if \(F_n\) is irreducible over \(\mathbb Q\) and \(n\) is even.
He is able to prove that \(F_n\) is irreducible over \(\mathbb Q\) if \(n=2p^k\), \(p\ge 5\) a prime or \(n=4p^k\), \(p\ge 11\) a prime.
These results are almost in the sense of I. Schur who gave the first families of rational polynomials whose Galois groups are \(A_n\) in cases \(n\equiv 0\pmod 4\) or \(n\equiv 1\pmod 2\).

Reviewer: Bernd Richter (Berlin)

MSC:

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

Keywords:

inverse Galois problem; generalized Laguerre polynomials; alternating group

References:

[1]	Grosswald, E., Bessel polynomials, (), No. 698 · Zbl 0188.09802
[2]	Schur, I., Gleichungen ohne affekt, (), 191-197 · JFM 56.0110.02
[3]	Schur, I., Affektlose gleichungen in der theorie der laguerreschen und hermiteschen polynome, (), 227-233 · JFM 57.0125.05