Some generalized Laguerre polynomials whose Galois groups are the alternating groups
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The Galois group of the generalized Laguerre polynomial F 2 n = e x x −2 n ( d 2 n ( e − x x 4 n )/ dx 2 n ) is the alternating group of degree 2 n, provided that F 2n is irreducible over the rationals as discussed by the authors.About:
This article is published in Journal of Number Theory.The article was published on 1989-02-01 and is currently open access. It has received 22 citations till now. The article focuses on the topics: Galois group & Generic polynomial.read more
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Algebraic Properties of a Family of Generalized Laguerre Polynomials
TL;DR: In this paper, the algebraic properties of generalized Laguerre polynomials for negative integral values of the parameter were studied and the conjecture was shown to be true when r is large with respect to n ≥ 5.
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On a certain family of generalized Laguerre polynomials
TL;DR: In this paper, the generalized Laguerre polynomial Ln(−3−n)(x)=∑j=0n1j!(n−j+1)(n− j+2)2xj is irreducible over the rationals for all n⩾1 and has Galois group An if n+1 is an odd square, and Sn otherwise.
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Algebraic properties of a family of Generalized Laguerre Polynomials
TL;DR: In this paper, the algebraic properties of generalized Laguerre polynomials for negative integral values of the parameter were studied for integers with constant n. The main tool is the theory of $p$-adic Newton Polygons.
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On the Galois group of generalized Laguerre polynomials
TL;DR: In this article, a simple criterion for the Galois group of a polynomial to be "large" was formulated using the theory of Newton polygons, and it was shown that for a fixed ε ∈ Q − Z < 0, Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L (�) n (x) = P n=0 n+� n j � (−x) j /j! is irreducible for all large enough n.
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Laguerre polynomials with Galois group Am for each m
TL;DR: Inverse Galois Theory as mentioned in this paper showed that for every positive integer m ≡ 2 ( mod 4 ), there is a Laguerre polynomial of degree m with associated Galois group A m.