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 in every solution (if any) of the equation $x^p + y^p + z^p = 0$ where $x, y$ and $z$ are $P$-integral elements of $L$ not divisible by $P,$ the rational integer $t$ congruent to $x/y$ (resp. $y/z,$ resp. $z/x$) modulo $P$ is a root of the Kummer-Mirimanoff system of congruences $$B_{2i}l^{p-2i}(t + \zeta) \equiv 0\pmod{p}$$ for $i = 1$ to $(p-3)/2$, and $l^{p-1}(t + \zeta) \equiv 0 \pmod{p}$, where $l^{j}$ denotes the $j$-th Kummer logarithmic function (with respect to $\zeta$).
Do you know if this was already published ?
Thanks in advance.
