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Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed $\alpha \in \mathbb{Q}-{\mathbb{Z}}_{\<0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre Polynomial $L_n^{\left(\alpha \right)}\left(x\right)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{n+\alpha }{n-j}\right){(-x)}^j/j!$ is irreducible for all large enough $n$. We use our criterion to show that, under these conditions, the Galois group of $L_n^{\left(\alpha \right)}\left(x\right)$ is either the alternating or symmetric group on $n$ letters, generalizing results of Schur for $\alpha =0,1,\pm \frac{1}{2},-1-n$.

Let $K$ be a number field, and suppose $\lambda (x,t)\in K[x,t]$ is irreducible over $K\left(t\right)$. Using algebraic geometry and group theory, we describe conditions under which the $K$-exceptional set of $\lambda$, i.e. the set of $\alpha \in K$ for which the specialized polynomial $\lambda (x,\alpha )$ is $K$-reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed $n\ge 10$, all but finitely many $K$-specializations of the degree $n$ generalized Laguerre polynomial $L_n^{\left(t\right)}\left(x\right)$ are $K$-irreducible and have Galois group $S_n$. Second, we study specializations...

The one-parameter family of polynomials $J_n(x,y)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{y+j}{j}\right)x^j$ is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each $n\ge 6$, the polynomial $J_n(x,y_0)$ is irreducible over $\mathbb{Q}$ for all but finitely many $y_0\in \mathbb{Q}$. If $n$ is odd, then with the exception of a finite set of $y_0$, the Galois group of $J_n(x,y_0)$ is $S_n$; if $n$ is even, then the exceptional set is thin.