Note on the ring of integers of a Kummer extension of prime degree. III

Abstract

Let $p$ be an odd prime number, $K$ a CM-field, and $K_{\infty}/K$ the cyclotomic $\mathbf{Z}_p$-extension with its $n$-th layer $K_n$ ($n \geq 0$). Let $A_n$ be the Sylow $p$-subgroup of the ideal class group of $K_n$. For the odd part $A_n^-$ of $A_n$, it is well known that the natural map $A_n^- \to A_{n+1}^-$ is injective. The purpose of this note is to show that an analogous phenomenon occurs for the Galois module structure of rings of integers of a certain class of tamely ramified extensions over $K_n$ of degree $p$.