I fell on the following fact : let p be an odd prime, let K denote the p-th cyclotomic field, let L be an extension of K with finite degree not divisible by p, and assume that the prime ideal $(1 - \zeta)$ of K (where $\zeta$ denotes a primitive p-th root of unity) ramifies completely in L. Let P denote the only prime ideal of L dividing $(1 - \zeta)$. Then the equation $x^p + y^p + z^p$ has no solution where x, y and z are P-integral elements of L not divisible by P. Do you know if it was already published ? Thanks in advance.