A remark on Greenberg's generalized conjecture for imaginary $S_3$-extensions of $\mathbb{Q}$

TL;DR

This paper provides a sufficient condition for Greenberg's generalized conjecture to hold in certain imaginary $S_3$-extensions of $Q$ where a prime splits into three primes, advancing understanding in algebraic number theory.

Contribution

It introduces a new sufficient condition for the validity of Greenberg's generalized conjecture in specific imaginary $S_3$-extensions of $Q$ with a prime splitting into three primes.

Findings

Established a sufficient condition for Greenberg's conjecture in these extensions.

Clarified the behavior of primes in $S_3$-extensions related to the conjecture.

Contributed to the theoretical understanding of Iwasawa theory in non-abelian extensions.

Abstract

Let $K/ \mathbb{Q}$ be an imaginary $S_3$-extension, and $p$ a prime number which splits into exactly three primes in $K$. We give a sufficient condition for the validity of Greenberg's generalized conjecture for $K$ and $p$.