On a characterization of algebraic number fields with class number less than three

Algebraic number fields

By an algebraic number field we mean a subfield of the algebraic numbers, or an isomorphic copy of such a field. Here we consider questions related to the complexity of determining isomorphism between algebraic number fields. We characterize the algebraic number fields with computable copies. For computable algebraic number fields, we give the complexity of the index sets. We show that the isom...

Normal Algebraic Number Fields.

Introduction. In this paper we present a detailed account of the results recently published in the Proceedings of the National Academy of Sciences [29 Our theory is an attempt to generalize the results of the classical class field theory to arbitrary normal fields. In the last analysis, the theory of cyclic extensions Z of an algebraic number field k can be described in terms of cyclic algebras...

On a Class Number Formula for Real Quadratic Number Fields

For an even Dirichlet character , we obtain a formula for L(1;) in terms of a sum of Dirichlet L-series evaluated at s = 2 and s = 3 and a rapidly convergent numerical series involving the central binomial coeecients. We then derive a class number formula for real quadratic number elds by taking L(s;) to be the quadratic L-series associated with these elds.

Isomorphisms of Algebraic Number Fields

Let Q(α) and Q(β) be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, Q(β) → Q(α). The algorithm is particularly efficient if there is only one isomorphism.

Class-number Problems for Cubic Number Fields