First-degree prime ideals of composite extensions

Let and be linearly disjoint number fields and let be their compositum. We prove that the first-degree prime ideals (FDPIs) of may almost always be constructed in terms of the FDPIs of and , and . We identify the cases where this correspondence does not hold, and provide explicit counterexamples for...

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Bibliographic Details
Published inJournal of mathematical cryptology Vol. 19; no. 1; pp. 50 - 94
Main Authors Santilli, Giordano, Taufer, Daniele
Format Journal Article
LanguageEnglish
Published Berlin De Gruyter 14.04.2025
Walter de Gruyter GmbH
Subjects
11Y05
11Y40
12F05
first-degree prime ideals
linearly disjoint extensions
Number theory
principal ideal factorization
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Summary:Let and be linearly disjoint number fields and let be their compositum. We prove that the first-degree prime ideals (FDPIs) of may almost always be constructed in terms of the FDPIs of and , and . We identify the cases where this correspondence does not hold, and provide explicit counterexamples for each obstruction. We show that for every pair of coprime integers , such a correspondence almost always respects the divisibility of principal ideals of the form , with a few exceptions that we characterize. Finally, we establish the asymptotic computational improvement of such an approach, and we verify the reduction in time needed for computing such primes for certain concrete cases.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:1862-2984
1862-2976
1862-2984
DOI:10.1515/jmc-2024-0036