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′.