On Fermat's equation over some quadratic imaginary number fields

TL;DR

Under a standard Langlands conjecture, the paper proves Fermat's Last Theorem over certain quadratic imaginary fields, extending the theorem's validity beyond rational numbers to specific quadratic fields.

Contribution

It demonstrates Fermat's Last Theorem over b1(i) and non-existence of solutions over b2(-2) and b2(-7) assuming a key Langlands conjecture, advancing understanding of Fermat's equation over quadratic fields.

Findings

Fermat's Last Theorem holds over b1(i) assuming the conjecture.

No non-trivial solutions for p e= 4 over b2(-2) and b2(-7) under the same assumption.

Extension of Fermat's theorem to specific quadratic imaginary 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 $p \geq 5$, Fermat's equation $a^p+b^p+c^p=0$ does not have non-trivial solutions over $\mathbb Q(\sqrt{-2})$ and $\mathbb Q(\sqrt{-7})$.