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On the irreducibility of a certain class of Laguerre polynomials. (English) Zbl 1019.11006

It is shown that almost all generalized Laguerre polynomials \(L_m^{(m)}\) are irreducible over the rationals. This implies, due to a result of R. Gow [J. Number Theory 31, 201-207 (1989; Zbl 0693.12009)], that for almost all even \(m\) the Galois group of these polynomials is the alternating group \(A_m\).

MSC:

11C08 Polynomials in number theory
12E05 Polynomials in general fields (irreducibility, etc.)
12F12 Inverse Galois theory
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
11R32 Galois theory

Citations:

Zbl 0693.12009
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Full Text: DOI

References:

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