2-ranks of class groups of Witt equivalent number fields

Abstract

In [CPS] we have observed that each class of Witt equivalent quadratic number fields, except for the singleton class containing only Q(yCT), con­tains a field whose class group has 2-rank as large as we wish. Here we generalize this o bservation from the case of quadratic number fields to fields of arbitrary e11en degree n. We prove that each class of Witt equiva­lent number fields of even degree n > 2 contains a field K with the 2-rank of class group as large as we wish. In fact, we prove a stronger result saying that the field in question has large 2-rank of S-class group for a finite set S of primes of K containing all infinite and all dyadic primes of the field. We combine here an interpretation of the parity of S-class numbers in terms of a localization map (Proposition 6) with a valuation-theoretic result of En­dler on the existence of fields with prescribed completions. The latter has been used in [Sz] to construct fields with prescribed Witt equivalence inva­riants. Here we discuss this technique again to make elear its applicability in constructing, in a given Witt class, number fields with special properties.