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

TL;DR

This paper explores properties of primes in cyclotomic fields related to a generalized Fermat conjecture, proposing conjectures that could simplify the problem to a more manageable form.

Contribution

It introduces new conjectures and prime decomposition properties in cyclotomic fields that relate to the SFLT2 conjecture, extending previous work by Gras and Qu extquoteright eme.

Findings

Prime decomposition properties in cyclotomic fields are characterized.

A weak conjecture is proposed that implies a simplified form of the SFLT2 equation.

A new conjecture is formulated that could imply the SFLT2 conjecture.

Abstract

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.