# The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields

@article{Freitas2015TheAF,
title={The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields},
author={Nuno Freitas and S. Siksek},
journal={Compositio Mathematica},
year={2015},
volume={151},
pages={1395 - 1415}
}
• Published 2015
• Mathematics
• Compositio Mathematica
Let $K$ be a totally real field. By the asymptotic Fermat’s Last Theorem over$K$ we mean the statement that there is a constant $B_{K}$ such that for any prime exponent $p>B_{K}$, the only solutions to the Fermat equation $$\begin{eqnarray}a^{p}+b^{p}+c^{p}=0,\quad a,b,c\in K\end{eqnarray}$$ are the trivial ones satisfying $abc=0$. With the help of modularity, level lowering and image-of-inertia comparisons, we give an algorithmically testable criterion which, if satisfied by $K$, implies the… Expand

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