On Fermat's equation over some quadratic imaginary number fields

Abstract

Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat's Last Theorem over $\mathbb{Q}(i)$. Under the same assumption, we also prove that, for all prime exponents $p\ge 5$, Fermat's equation $a^p+b^p+c^p=0$ does not have non-trivial solutions over $\mathbb{Q}(\sqrt{-2})$ and $\mathbb{Q}(\sqrt{-7})$.

arXiv commentThe present is a revised version, including suggestions from referees, that was accepted for publication in Research in Number Theory; 16 pages

References in corpus (2)

• The Asymptotic Fermat's Last Theorem for Five-Sixths of Real Quadratic Fields
• Fermat's Last Theorem over some small real quadratic fields

Cited by in corpus (2)

• On Serre's modularity conjecture and Fermat's equation over quadratic imaginary fields of class number one
• Irreducibility of mod p Galois representations of elliptic curves with multiplicative reduction over number fields