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

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 \(n\equiv 1\pmod 2\).


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


[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
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