Sum-product theorems in algebraic number fields

Sum-product Theorems in Algebraic Number Fields

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...

Some Problems Related to Sum-product Theorems

The study of sum-product phenomena and product phenomena is an emerging research direction in combinatorial number theory that has already produced several striking results. Many related problems are not yet fully understood, or are far from being resolved. In what follows we propose several questions where progress can be expected and should lead to advances in this general area. For two finit...

Sum-product Theorems and Incidence Geometry

In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P1, · · · , P4, and Q1, · · · , Qn ∈ C, if there are ≤ n 1+δ 2 many distinct lines between Pi and Qj for all i, j, then P1, · · · , P4 are collinear. If the number of the distinct lines is < cn 1 2 , then the cross ratio of the four points is algebraic. 2. Given c > 0, there is δ > 0 such th...