First-degree prime ideals of composite extensions
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913793436549120 |
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| author | Santilli, Giordano Taufer, Daniele |
| author_facet | Santilli, Giordano Taufer, Daniele |
| contents | Let $\mathbb{Q}(α)$ and $\mathbb{Q}(β)$ be linearly disjoint number fields and let $\mathbb{Q}(θ)$ be their compositum. We prove that the first-degree prime ideals of $\mathbb{Z}[θ]$ may almost always be constructed in terms of the first-degree prime ideals of $\mathbb{Z}[α]$ and $\mathbb{Z}[β]$, and vice-versa. We also classify the cases in which this correspondence does not hold, by providing explicit counterexamples. We show that for every pair of coprime integers $d,e \in \mathbb{Z}$, such a correspondence almost always respects the divisibility of principal ideals of the form $(e+dθ)\mathbb{Z}[θ]$, with a few exceptions that we characterize. Finally, we discuss the computational improvement of such an approach, and we verify the reduction in time needed for computing such primes for certain concrete cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_04660 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | First-degree prime ideals of composite extensions Santilli, Giordano Taufer, Daniele Number Theory 11Y05, 11Y40, 12F05 Let $\mathbb{Q}(α)$ and $\mathbb{Q}(β)$ be linearly disjoint number fields and let $\mathbb{Q}(θ)$ be their compositum. We prove that the first-degree prime ideals of $\mathbb{Z}[θ]$ may almost always be constructed in terms of the first-degree prime ideals of $\mathbb{Z}[α]$ and $\mathbb{Z}[β]$, and vice-versa. We also classify the cases in which this correspondence does not hold, by providing explicit counterexamples. We show that for every pair of coprime integers $d,e \in \mathbb{Z}$, such a correspondence almost always respects the divisibility of principal ideals of the form $(e+dθ)\mathbb{Z}[θ]$, with a few exceptions that we characterize. Finally, we discuss the computational improvement of such an approach, and we verify the reduction in time needed for computing such primes for certain concrete cases. |
| title | First-degree prime ideals of composite extensions |
| topic | Number Theory 11Y05, 11Y40, 12F05 |
| url | https://arxiv.org/abs/2111.04660 |