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

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 $p$.