On Furtw\"angler's theorems and second case of Fermat's Last Theorem

TL;DR

This paper generalizes Furtw"angler's theorems for the second case of Fermat's Last Theorem, relating prime divisors of certain expressions to class group properties and $p$-power residue symbols, assuming Vandiver's conjecture.

Contribution

It extends Furtw"angler's theorems to the second case of FLT, establishing new relations between prime divisors, class groups, and residue symbols in cyclotomic fields.

Findings

If FLT2 fails, certain prime divisors are $p$-principal under Vandiver's conjecture.

Explicit formulas for $p$-power residue symbols of cyclotomic units.

Relations between prime divisors of specific expressions and class group properties.

Abstract

This article, complement to the article [Que], deals with some generalizations of Futw\"angler's theorems for the second case of Fermat's Last Theorem (FLT2). Let $p$ be an odd prime, $\zeta$ a $p$th primitive root of unity, $K:=\Q(\zeta)$ and $C\ell_K$ the class group of $K$. A prime $q$ is said $p$-principal if the class $c\ell_K (\mk q_K)\in C\ell_K$ of any prime ideal $\mk q_K$ of $\Z_K$ over $q$ is the $p$th power of a class. Assume that FLT2 fails for $(p,x,y,z)$ where $x, y, z$ are mutually coprime integers, $p$ divides $y$ and $x^p+y^p+z^p=0$. Let $q$ be a prime dividing $\frac{(x^p+y^p)(y^p+z^p)(z^p+x^p)}{(x+y)(y+z)(z+x)}$ and $\mk q_K$ be any prime ideal of $K$ over $q$. We obtain the $p$-power residue symbols relations: $$(\frac{p}{\mk q_K})_K=(\frac{1-\zeta^j}{\mk q_K})_K for j=1,\dots,p-1.$$ As an application, we prove that: if Vandiver's conjecture holds for $p$ then $q$ is a $p$-principal prime. Similarly, let $q$ be a prime dividing $\frac{(x^p-y^p)(y^p-z^p)(z^p-x^p)}{(x-y)(y-z)(z-x)}$ and $\mk q_K$ be the prime ideal of $K$ over $q$ dividing $(x\zeta-y)(z\zeta-y)(x\zeta -z)$. We give an explicit formula for the $p$-power residue symbols $(\frac{\epsilon_{k}}{\mk q_K})_K$ for all $k$ with $1<k\leq\frac{p-1}{2},$ where $\epsilon_k$ is the cyclotomic unit given by $\epsilon_k=:\zeta^{(1-k)/2}\cdot\frac{1+\zeta^k}{1+\zeta}.$ The principle of proofs rely on the $p$-Hilbert class field theory.