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

MSC:

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

[1] Grosswald, E., Bessel polynomials, (Lecture Notes in Mathematics (1978), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York), No. 698 · Zbl 0188.09802
[2] Schur, I., Gleichungen ohne Affekt, (Gesammelte Abhandlungen, Vol. 3 (1973), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York), 191-197
[3] Schur, I., Affektlose Gleichungen in der Theorie der Laguerreschen und Hermiteschen Polynome, (Gesammelte Abhandlungen, Vol. 3 (1973), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York), 227-233
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