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, $K:=\text{Q}(\zeta )$ the $p$th cyclotomic field and ${\mathbb{Z}}_K$ its ring of integers. The diophantine equation $(u+v\zeta ){\mathbb{Z}}_K=\text{mk}{w}_1^p$, with $u,v\in \mathbb{Z}\mathrm{\backslash }\{0\}$ coprime, $uv\equiv 0p$ and $\text{mk}{w}_1$ ideal of ${\mathbb{Z}}_K$, has no solution.} Assuming that $SFLT2$ fails for $(p,u,v)$, let $q$ be an odd prime not dividing $uv$, $n$ the order of $\frac{v}{u}q$, $\xi$ a primitive $n$th root of unity and $M:=\text{Q}(\xi ,\zeta )$. 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 $\text{mk}Q$ of ${\mathbb{Z}}_M$ 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.