On a certain family of generalized Laguerre polynomials. (English) Zbl 1053.11083

Sell, Elizabeth A.

J. Number Theory 107, No. 2, 266-281 (2004).

From the abstract: Following the work of Schur and Coleman, we prove the generalized Laguerre polynomial \(L_n^{(-3-n)}(x)=\sum_{j=0}^n \frac{1}{j!}\frac{(n-j+1)(n-j+2)}{2}x^j \) is irreducible over the rationals for all \(n\geq 1\) and has Galois group \(A_n\) if \(n+1\) is an odd square, and \(S_n\) otherwise. We also show that for certain negative integer values of \(\alpha\), Greek and certain congruence classes of \(n\) modulo 8, the splitting field of \(L_n^{\alpha}(x)\) can be embedded in a double cover.

Reviewer: Tanguy Rivoal (Caen)

MSC:

11R09	Polynomials (irreducibility, etc.)
11C08	Polynomials in number theory
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

Generalized Laguerre polynomials; Newton polygons

References:

[1]	Chebyshev, P.I, Sur la totalité des nombres premiers inférieurs à une limite donnée, J. math., 17, 341-365, (1852)
[2]	Coleman, R.F, On the Galois groups of the exponential Taylor polynomials, L’enseignement math., 33, 183-189, (1987) · Zbl 0672.12004
[3]	Dumas, G, Sur quelques cas d’irréducibilité des polynomes à coefficients rationnels, J. math. pures appl., 2, 191-258, (1906) · JFM 37.0096.01
[4]	Feit, W, \(Ã5\) and \(Ã7\) are Galois groups over number fields, J. algebra, 104, 231-260, (1986)
[5]	M. Filaseta, A generalization of an irreducibility theorem of I. Schur, in: B.C. Berndt, H.G. Diamond, A.J. Hildebrand (Eds.), Analytic Number Theory, Vol. 1, Progr. Math. 138. · Zbl 0854.11052
[6]	Filaseta, M; Lam, T.-Y, On the irreducibility of the generalized Laguerre polynomials, Acta arith., 105, 2, 177-182, (2002) · Zbl 1010.12001
[7]	Gow, R, Some generalized Laguerre polynomials whose Galois groups are the alternating groups, J. number theory, 31, 201-207, (1989) · Zbl 0693.12009
[8]	Hajir, F, Some \(Añ\)-extensions obtained from generalized Laguerre polynomials, J. number theory, 50, 206-212, (1995) · Zbl 0829.12004
[9]	Jordan, C, Traité des substitutions et des équations algébriques, (1870), Gauthier-Villars Paris
[10]	Jordan, C, Sur la limite de transitivité des groupes non alternés, Bull. soc. math. France, 1, 40-71, (1872-1873)
[11]	Karpilovsky, G, Projective representations of finite groups, (1985), Marcel Dekker New York · Zbl 0571.20004
[12]	Koblitz, N, P-adic numbers, p-adic analysis, and zeta-functions, (1984), Springer Berlin
[13]	Ledet, A, On a theorem by Serre, Proc. AMS, 128, 27-29, (1999) · Zbl 0940.12002
[14]	Pólya, G; Szegő, G, Problems and theorems in analysis II, (1976), Springer Berlin · Zbl 0311.00002
[15]	I. Schur, Gemsammelte Abhandlungen, Vol. 3, Springer, Berlin, 1973.
[16]	Serre, J.-P, A course in arithmetic, (1973), Springer Berlin
[17]	Serre, J.-P, L’invariant De Witt de la forme tr(x2), Comment. math. helv., 59, 651-676, (1984) · Zbl 0565.12014
[18]	Weiss, E, Algebraic number theory, (1963), Chelsea London
[19]	E.A. Wrobel, On a Certain Class of Generalized Laguerre Polynomials, Master’s Thesis, University of North Carolina at Chapel Hill, 2003.