On smooth ideals in number fields
type: Item
instance of: scholarly article
title: On smooth ideals in number fields
zbMATH Open document ID: 0859.11060
author: Christine S. Hollinger
author: Johannes Buchmann
published in: Journal of Number Theory
publication date: 1996-11-10T00:00:00Z
review text: The authors find a lower bound for \(\psi_F(x,y)\), the number of integral ideals in \(\mathbb{F}\) of norm \(\leq x\) which are \(y\)-smooth (have all prime factors of norm \(\leq y)\). The rational result, (needed for primality testing), is essentially that \(\log (\psi_\mathbb{Q} (x,y)/x)\) has order at least \(\approx \log (1/u^u)\) for large \(u= \log x/ \log y\) [see \textit{C. Pomerance}, Math. Cent. Tracts 154, 89-139 (1982; Zbl 0508.10004)]. The generalized result for an arbitrary field is much the same in magnitude, but it is bolstered with precise orders of magnitude of errors. The estimate is simplified by restriction to the case of prime ideals which split completely in the closure of \(\mathbb{F}/ \mathbb{Q}\). The GRH is invoked for the error estimates on \(\pi(w) - \text{Li}(w)/n\) with splitting primes in a normal field of degree \(n\) [\textit{J. Oesterlé}, Astérisque 61, 165-167 (1979; Zbl 0418.12005)].
reviewed by: Harvey Cohn
Mathematics Subject Classification ID: 11R44
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 11R45
zbMATH DE Number: 913831
zbMATH Keywords: lower bound for the number of smooth integral ideals
zbMATH Keywords: error term
zbMATH Keywords: primality testing
zbMATH Keywords: splitting primes
MaRDI profile type: Publication
OpenAlex ID: W2044530517
DOI: 10.1006/JNTH.1996.0088
description: scientific article; zbMATH DE number 913831