Galois groups of (ⁿ₀)+(ⁿ₁)X+…+(ⁿ₆)X⁶

We show that the Galois group of the polynomial in the title is isomorphic to the full symmetric group on six symbols for all but finitely many n. This complements earlier work of Filaseta and Moy, who studied Galois groups of (Formular Presented) for more general pairs (n, k), but had to admit a possibly infinite exceptional set specifically for k = 6 of at most logarithmic growth in n. The proof rests upon invoking Faltings’ theorem on a suitable fibration of Galois resolvents to show that this exceptional set is, in fact, finite.