Wild sets and 2-ranks of class groups

P. Conner, R. Perlis, K. Szymiczek
1997 Acta Arithmetica  
1. Introduction. The notion of two number fields K and L having equivalent Hilbert-symbol reciprocity laws was introduced in [PSCL] for the purpose of studying isomorphisms between the Witt rings W (K) and W (L). In that paper, the name reciprocity equivalence was used; however, in view of recent developments in which equivalences between higher-order reciprocity laws are discussed (see [CzSł]), these earlier equivalences are now more properly referred to as Hilbert-symbol equivalences. Here is
more » ... uivalences. Here is the definition: A Hilbert-symbol equivalence between K and L is a pair of maps (t, T ) in which t : K * /K * 2 → L * /L * 2 is an isomorphism of square-class groups, and T : Ω K → Ω L is a bijection between the set of places of K and those of L, preserving Hilbert-symbols in the sense that (a, b) P = (ta, tb) T P for all square-classes a, b ∈ K * /K * 2 and all places P of K. For the convenience of the reader, we recall that there is a Hilbert-symbol equivalence between number fields K and L if and only if K and L have the same level, the same number of real infinite places, and if there is a bijection between the dyadic places of K and of L so that the corresponding dyadic completions have the same degree over Q 2 and also have the same local level. For details, see Theorem 1.5 of [Szym].
doi:10.4064/aa-79-1-83-91 fatcat:o6oe3vspqfhlzesdu73dxepfnu