Leopoldt-type theorems for non-abelian extensions of

Abstract

Abstract We prove new results concerning the additive Galois module structure of wildly ramified non-abelian extensions $K/\mathbb{Q}$ with Galois group isomorphic to $A_4$ , $S_4$ , $A_5$ , and dihedral groups of order $2p^n$ for certain prime powers $p^n$ . In particular, when $K/\mathbb{Q}$ is a Galois extension with Galois group $G$ isomorphic to $A_4$ , $S_4$ or $A_5$ , we give necessary and sufficient conditions for the ring of integers $\mathcal{O}_{K}$ to be free over its associated order in the rational group algebra $\mathbb{Q}[G]$ .