Galois groups of \binomn0 + \binomn1 X + … + \binomn6 X⁶

2023/04/25 by Benjamin Klahn, Klahn, Benjamin, Marc Technau +1
Mathematics · #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems #History and Theory of Mathematics

paper · pdf · doi:10.48550/arxiv.2304.12658

Abstract

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 \binomn0 + \binomn1 X + … + \binomnk 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.

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