Approximating rings of integers in number fields

TLDR: This paper studies the algorithmic problem of finding the ring of integers of a given algebraic mimber field and proves that this subring has a particularly transparent local structure, which is reminiscent of the structure of tamely ramified extensions of local fields.

Abstract: In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic mimber field. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for facto- ring integers. Applying a variar.t of a Standard algorithm for finding rings of integers, one finds a subring of the number field that one may view as the "best guess" one has for the ring of integers. This best guess is probably often correct. Our main cor.cern is what can be proved about this subring. We show that it has a particularly transparent local structure, which is reminiscent of the structure of tamely ramified extensions of local fields. A major portion of the paper is devoted to the study of rings that are "tarne" in our more general sense. As a byproduct, we prove complexity results that elaborate upon a result of Chistov. The paper also includes a section that discusses polynomial time algorithms related to finitely generated abelian groups.