Complements on Furtw\"angler's second theorem and Vandiver' s cyclotomic integers

This article deals with a conjecture generalizing the second case of Fermat's Last Theorem, called $SFLT2$ conjecture: {\it Let $p>3$ be a prime, $K:=\Q(\zeta)$ the $p$th cyclotomic field and $\Z_K$ its ring of integers. The diophantine equation $(u+v\zeta)\Z_K=\mk w_1^p$, with $u,v\in\Z\backslash\{0\}$ coprime, $uv\equiv 0 \bmod p$ and $\mk w_1$ ideal of $\Z_K$, has no solution.} Assuming that $SFLT2$ fails for $(p,u,v)$, let $q$ be an odd prime not dividing $uv$, $n$ the order of $\frac{v}{u}\bmod q$, $\xi$ a primitive $n$th root of unity and $M:=\Q(\xi,\zeta)$. The aim of this complement of the article [GQ] of G. Gras and R. Qu\^eme on the same topic, is to exhibit some strong properties of the decomposition of the primes $\mk Q$ of $\Z_M$ over $q$ in certain Kummer $p$-extensions of the field $M$, to derive from them a weak conjecture which implies that the SFLT2 equation can always take the reduced form $u+\zeta v\in K^{\times p}$ and to set a conjecture implying SFLT2.