An equivalence between local fields

The $p$-component of the index of a number field $K$ depends only on the completions of $K$ at the primes over $p$. In this paper we define an equivalence relation between $m$-tuples of local fields such that, if two number fields $K$ and $K'$ have equivalent $m$-tuples of completions at the primes over $p$, then they have the same $p$-component of the index. This equivalence can be interpreted in terms of the decomposition groups of the primes over $p$ of the normal closures of $K$ and $K'$.