Vol. 9, No. 4, 2015

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Fermat's last theorem over some small real quadratic fields

Nuno Freitas and Samir Siksek

Vol. 9 (2015), No. 4, 875–895
Abstract

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations, and ray class groups, we show that for $3\le d\le 23$, where $d\ne 5,17$ and is squarefree, the Fermat equation ${x}^{n}+{y}^{n}={z}^{n}$ has no nontrivial solutions over the quadratic field $ℚ\left(\sqrt{d}\right)$ for $n\ge 4$. Furthermore, we show that for $d=17$, the same holds for prime exponents $n\equiv 3,5\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}8\right)$.

Keywords
Fermat, modularity, Galois representation, level lowering
Mathematical Subject Classification 2010
Primary: 11D41
Secondary: 11F80, 11F03