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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 $B_{2i}l^{p-2i}(t + \zeta) \equiv 0$ (mod 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.

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.

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 $B_{2i}l^{p-2i}(t + \zeta) \equiv 0$ (mod 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 espect to $\zeta$). Do you know if this was already published ? Thanks in advance.

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 $B_{2i}l^{p-2i}(t + \zeta) \equiv 0$ (mod 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.

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 $B_{2i}l^{p-2i}(t + \zeta) \equiv 0$ (mod p) for i = 1 to (p-3)/2 and $l^{p-1}(t + \zeta) \equiv 0$ (mod p). Do you know if this was already published ? Thanks in advance.

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 $B_{2i}l^{p-2i}(t + \zeta) \equiv 0$ (mod 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 espect to $\zeta$). Do you know if this was already published ? Thanks in advance.