On Ore’s conjecture and its developments

Abstract:

The $p$-component of the index of a number field $K$, ${ \rm ind}_p(K)$, depends only on the completions of $K$ at the primes over $p$. More precisely, $\textrm {ind}_p(K)$ equals the index of the $\mathbb {Q}_p$-algebra $K\otimes \mathbb {Q}_p$. If $K$ is normal, then $K\otimes \mathbb {Q}_p\cong L^n$ for some $L$ normal over $\mathbb {Q}_p$ and some $n$, and we write $I_p(nL)$ for its index. In this paper we describe an effective procedure to compute $I_p(nL)$ for all $n$ and all normal and tamely ramified extensions $L$ of $\mathbb {Q}_p$, hence to determine $\textrm {ind}_p(K)$ for all Galois number fields that are tamely ramified at $p$. Using our procedure, we are able to exhibit a counterexample to a conjecture of Nart (1985) on the behaviour of $I_p(nL)$.