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It is shown that the quartic Fermat equation $x^4 +y^4=1$ has nontrivialintegral solutions in the Hilbert class field $\Sigma$ of any quadratic field$K=\mathbb{Q}(\sqrt{-d})$ whose discriminant satisfies $-d \equiv 1$ (mod 8). Acorollary is that the quartic Fermat equation has no nontrivial solution in$K=\mathbb{Q}(\sqrt{-p})$, for $p$ $( > 7)$ a prime congruent to $7$ (mod 8),but does have a nontrivial solution in the odd degree extension $\Sigma$ of$K$. These solutions arise from explicit formulas for the points of order 4 onelliptic curves in Tate normal form. The solutions are studied in detail andthe results are applied to prove several properties of the Weber singularmoduli introduced by Yui and Zagier.

The quartic Fermat equation in Hilbert class fields of imaginary quadratic fields