Local criteria for the unit equation and the asymptotic Fermat's Last Theorem.

Freitas, Nuno; Kraus, Alain; Siksek, Samir · Proc Natl Acad Sci U S A · 2021

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Abstract

Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat's Last Theorem to hold over F and also, for the nonexistence of solutions to the unit equation over F. For example, if two totally ramifies and three splits completely in F, then the asymptotic Fermat's Last Theorem holds over F.