Density results for the 2-classgroups of imaginary quadratic fields

Abstract

In one of the long series of papers, Rédie [15] has given a theoretical description of the first three ‘levels’ of the 2-classgroup of a quadratic number field. Starting from Reichardt’s characterization [18] of the 2^n-rank (which we done by e_(2n)) of the restriced classgroup L of the field Q(1√4), Rédie characterized e_4 and e_8 in terms of certain factorizations of Δ and a 2-valued multiplicative symbol {a_1, a_2, a_3}. This symbol is closely related to the splitting of primes in eight degree extensions of Q (see [2]). Using this symbol, Rédie was able to prove that there are infinitely many real quadratic fields for which e_2, e_4, and e_8 have arbitrarily assigned values.