On characterization of Monogenic number fields associated with certain quadrinomials and its applications
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915093241921536 |
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| author | Chatterjee, Tapas Kumar, Karishan |
| author_facet | Chatterjee, Tapas Kumar, Karishan |
| contents | Let $f(x)=x^{n}+ax^{3}+bx+c$ be the minimal polynomial of an algebraic integer $θ$ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ Let $K=\mathbb{Q}(θ)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $θ.$ As an interesting result, we establish necessary and sufficient conditions for the field $K=\mathbb{Q}(θ)$ to be monogenic. Finally, we investigate the types of solutions to certain differential equations associated with the polynomial $f(x).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14117 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On characterization of Monogenic number fields associated with certain quadrinomials and its applications Chatterjee, Tapas Kumar, Karishan Number Theory Primary: 11R04, 11R29, 11Y40, Secondary: 11R09, 11R21 Let $f(x)=x^{n}+ax^{3}+bx+c$ be the minimal polynomial of an algebraic integer $θ$ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ Let $K=\mathbb{Q}(θ)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we characterize all the prime divisors of the discriminant of $f(x)$ which do not divide the index of $θ.$ As an interesting result, we establish necessary and sufficient conditions for the field $K=\mathbb{Q}(θ)$ to be monogenic. Finally, we investigate the types of solutions to certain differential equations associated with the polynomial $f(x).$ |
| title | On characterization of Monogenic number fields associated with certain quadrinomials and its applications |
| topic | Number Theory Primary: 11R04, 11R29, 11Y40, Secondary: 11R09, 11R21 |
| url | https://arxiv.org/abs/2408.14117 |