On Galois groups of generalized Laguerre polynomials whose discriminants are squares
The author computes the Galois group of Laguerre polynomials NEWLINE\[NEWLINEL_n^{(\alpha)}(x)=\sum_{j=0}^n \dfrac{(n+\alpha)(n-1+\alpha) \cdots (j+1+\alpha)}{(n-j)!j!} (-x)^j,NEWLINE\]NEWLINE with \(n\) and \(\alpha\) integer numbers, whose discriminant is a square. In [the author et al., J. Théor. Nombres Bordx. 25, No. 1, 1--30 (2013; Zbl 1287.11128)], an explicit description of the integer pairs \((n,\alpha)\) for which the discriminant of \(L_n^{(\alpha)}(x)\) is a rational square was obtained.NEWLINENEWLINEIn the paper under review, it is shown that, among these pairs, for \(2\leq n \leq 5\), the Galois group of \(L_n^{(\alpha)}(x)\) is the alternating group \(A_n\) except for the pairs \((4,-1)\) and \((4,23)\). For \(n\geq 6\), it is shown that the Galois group of \(L_n^{(\alpha)}(x)\) is \(A_n\) if and only if \(L_n^{(\alpha)}(x)\) is irreducible over the field of rational numbers. The proof uses the Newton polygons of the \(L_n^{(\alpha)}(x)\) as well as results of \textit{G. Dumas} [Journ. de Math. (6) 2, 191--258 (1906; JFM 37.0096.01)] to establish irreducibility, and of \textit{F. Hajir} [J. Théor. Nombres Bordx. 17, No. 2, 517--525 (2005; Zbl 1094.11042)] to determine the Galois groups. For the pairs \((n,\alpha)\) for which the discriminant of \(L_n^{(\alpha)}(x)\) is a rational square and \(n\leq 35\), the author has checked, using Sage, that the Galois group of \(L_n^{(\alpha)}(x)\) is \(A_n\), either directly or checking that \(L_n^{(\alpha)}(x)\) is irreducible.
scientific article; zbMATH DE number 6335258
scientific article