2-torsion of class groups of multiquadratic fields

The narrow class group of a number field is one of the most fundamental and, yet mysterious, objects in arithmetic. Its study was initiated by Gauss in 1801, using the language of quadratic forms. In his dissertation Gauss reported what still is one of the very few explicit results about the class group, namely an explicit description of the 2-torsion of the class group and the dual class group of a quadratic number field.

         In this talk I will explain a recent joint work with Peter Koymans, where we have extended this result to all multi-quadratic number fields. We provide a sharp upper bound for the size of the 2-torsion of the class group of a multi quadratic field K and an explicit description of H(K,2)/K, the maximal unramified exponent 2 extension of K. To do so we introduce and closely describe expansion groups and expansion Lie algebras in terms of universal objects. It turns out that the 2-torsion of the narrow class group of K reaches the bound if and only if the Galois group Gal(H(K,2)/Q) is universal. Finally, for each positive integer n, we combine this description with Ramsey theory to find infinitely many K with Gal(H(K,2)/Q) universal and Gal(K/Q) being of dimension n over the field of 2 elements. These ideas are inspired by Alexander Smith's recent breakthrough on the Goldfeld's and Cohen--Lenstra conjectures.