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We answer three reducibility (or irreducibility) questions for $0,1$-polynomials, those polynomials which have every coefficient either $0$ or $1$. The first concerns whether a naturally occurring sequence of reducible polynomials is finite. The second is whether every nonempty finite subset of an infinite set of positive integers can be the set of positive exponents of a reducible $0,1$-polynomial. The third is the analogous question for exponents of irreducible $0,1$-polynomials.

We show that the discriminant of the generalized Laguerre polynomial $L_n^{\left(\alpha \right)}\left(x\right)$ is a non-zero square for some integer pair $(n,\alpha )$, with $n\ge 1$, if and only if $(n,\alpha )$ 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 $L_n^{\left(\alpha \right)}\left(x\right)$ over $\mathbb{Q}$ is the alternating group $A_n$. For example, we establish that for all but finitely many positive integers $n\equiv 2(mod4)$, the only $\alpha$ for which the Galois group of $L_n^{\left(\alpha \right)}\left(x\right)$ over $\mathbb{Q}$ is $A_n$ is $\alpha =n$.