Approximating Rings of Integers in Number Elds Approximating Rings of Integers in Number Elds

In this paper we study the algorithmic problem of nding the ring of integers of a given algebraic number eld. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number elds that are deened by equations with very large coeecients. Such elds occur in the number eld sieve algorithm for factoring integers. Applying a variant of a standard algorithm for nding rings of integers, one nds a subring of the number eld that one may view as the \best guess" one has for the ring of integers. This best guess is probably often correct. Our main concern 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 ramiied extensions of local elds. A major portion of the paper is devoted to the study of rings that are \tame" 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 nitely generated abelian groups.