2-ranks of class groups of Witt equivalent number fields - Szymiczek, Kazimierz (Annales Mathematicae Silesianae, 1998)
In [CPS] we have observed that each class of Witt equivalent quadratic number fields, except for the singleton class containing only ℚ(√-l), contains a field whose class group has 2-rank as large as we wish. Here we generalize this observation from the case of quadratic number fields to fields of arbitrary even 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 clear its applicability in constructing, in a given Witt class, number fields with special properties.