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On classifying Laguerre polynomials which have Galois group the alternating group

Pradipto Banerjee; Michael Filaseta; Carrie E. Finch; J. Russell Leidy — 2013

Journal de Théorie des Nombres de Bordeaux

We show that the discriminant of the generalized Laguerre polynomial $L_{n}^{(α)} (x)$ is a non-zero square for some integer pair $(n, α)$ , with $n \geq 1$ , if and only if $(n, α)$ belongs to one of $30$ explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of $L_{n}^{(α)} (x)$ over $ℚ$ is the alternating group $A_{n}$ . For example, we establish that for all but finitely many positive integers $n \equiv 2 (mod 4)$ , the only $α$ for which the Galois group of $L_{n}^{(α)} (x)$ over $ℚ$ is $A_{n}$ is $α = n$ .

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