Galois groups of $\binom{n}{0} + \binom{n}{1} X + \ldots + \binom{n}{6} X^6$

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 $(\binom{n}{0}) + (\binom{n}{1}) X + \dots + (\binom{n}{k}) X^{k}$ 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.

Galois groups of $(\binom{n}{0}) + (\binom{n}{1}) X + \dots + (\binom{n}{6}) X^{6}$

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Galois groups of $(\binom{n}{0}) + (\binom{n}{1}) X + \dots + (\binom{n}{6}) X^{6}$

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Galois groups of (n0)+(n1)X+…+(n6)X6(0n​)+(1n​)X+…+(6n​)X6

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Galois groups of (n0)+(n1)X+…+(n6)X6(0n​)+(1n​)X+…+(6n​)X6

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Galois groups of $(\binom{n}{0}) + (\binom{n}{1}) X + \dots + (\binom{n}{6}) X^{6}$

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Galois groups of $(\binom{n}{0}) + (\binom{n}{1}) X + \dots + (\binom{n}{6}) X^{6}$

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