On number fields with an unramified abelian extension of degree \(2^{n+2}\)
Let \(K\) be either a totally real or a totally imaginary number field. The author considers a special family of cubic polynomials \(f_ \lambda(x)\in K[x]\), each of which generates a cubic cyclic extension \(L\) of \(K\). For each \(L\) he is able to construct an unramified abelian extension of degree \(2^{n+2}\), where \(2^ n\) denotes the degree of the maximal unramified abelian 2-extension of \(K\). He makes the family of \(f_ \lambda(x)\) more explicit in the case that \(K\) is a quadratic field. In this case there is also a well-known lower bound for \(n\) coming from genus theory.
scientific article; zbMATH DE number 205451