An unsolved question surrounding the Generalized Laguerre Polynomial $$L_{n}^{(n)}(x)$$

We examine the family of generalized Laguerre polynomials \(L_{n}^{(n)}(x)\). In 1989, Gow discovered that if n is even, then the discriminant of \(L_{n}^{(n)}(x)\) is a nonzero square of a rational number. Additionally, in the case where the polynomial \(L_{n}^{(n)}(x)\) is irreducible over the rationals, the associated Galois group is the alternating group \(A_{n}\). Filaseta et al. (2012) established the irreducibility of \(L_{n}^{(n)}(x)\) for every \(n>2\) satisfying \(2\pmod {4}\). They also demonstrated that if n is \(0\pmod {4}\), then \(L_{n}^{(n)}(x)\) has a linear factor if it is not irreducible. The question of whether \(L_{n}^{(n)}(x)\) has a linear factor when n is \(0\pmod {4}\) remained unanswered. We resolve this question by proving that \(L_{n}^{(n)}(x)\) does not have a linear factor for sufficiently large n. This conclusion completes the classification of generalized Laguerre polynomials having Galois group the alternating group, excluding a finite set of exceptions.