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pro vyhledávání: '"11R04, 11R29"'
Number fields and their rings of integers, which generalize the rational numbers and the integers, are foundational objects in number theory. There are several computer algebra systems and databases concerned with the computational aspects of these.
Externí odkaz:
http://arxiv.org/abs/2409.18030
Autor:
Chatterjee, Tapas, Kumar, Karishan
Let $\theta$ be an algebraic integer and $f(x)=x^{n}+ax^{n-1}+bx+c$ be the minimal polynomial of $\theta$ over the rationals. Let $K=\mathbb{Q}(\theta)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K.$ In this article, we chara
Externí odkaz:
http://arxiv.org/abs/2408.14524
Autor:
Chatterjee, Tapas, Kumar, Karishan
Let $f(x)=x^{n}+ax^{3}+bx+c$ be the minimal polynomial of an algebraic integer $\theta$ over the rationals with certain conditions on $a,~b,~c,$ and $n.$ Let $K=\mathbb{Q}(\theta)$ be a number field and $\mathcal{O}_{K}$ be the ring of integers of $K
Externí odkaz:
http://arxiv.org/abs/2408.14117
Let $\theta$ be a root of a monic polynomial $h(x) \in \Z[x]$ of degree $n \geq 2$. We say $h(x)$ is monogenic if it is irreducible over $\Q$ and $\{ 1, \theta, \theta^2, \ldots, \theta^{n-1} \}$ is a basis for the ring $\Z_K$ of integers of $K = \Q(
Externí odkaz:
http://arxiv.org/abs/2402.10131
Autor:
Grenié, Loïc, Molteni, Giuseppe
Under the assumption of the validity of the Generalized Riemann Hypothesis, we prove that the class group of every field of degree $n$ and discriminant with absolute value $\Delta$ can be generated using prime ideals with norm $\leq (4-1/(2n))\log^2\
Externí odkaz:
http://arxiv.org/abs/2212.09461
Autor:
Battistoni, Francesco, Zaïmi, Toufik
Let $Tr$ denote the trace $\mathbb{Z}$-module homomorphism defined on the ring $\mathcal{O}_{L} $ of the integers of a number field $L.$ We show that $Tr(\mathcal{O}_{L})\varsubsetneq \mathbb{Z}$ if and only if there is a prime factor $p$ of the degr
Externí odkaz:
http://arxiv.org/abs/2110.06614
Autor:
Kaur, Sumandeep, Khanduja, Sudesh Kaur
Let $K=\mathbb Q(\theta)$ be an algebraic number field with $\theta$ a root of an irreducible trinomial $f(x)=x^6+ax+b$ belonging to $\mathbb{Z}[x]$. In this paper, for each prime number $p$ we compute the highest power of $p$ dividing the discrimina
Externí odkaz:
http://arxiv.org/abs/2011.14348
Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this
Externí odkaz:
http://arxiv.org/abs/2005.01915
Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this
Externí odkaz:
http://arxiv.org/abs/2005.01300
Autor:
Carlson, Magnus, Kim, Minhyong
We compute some arithmetic path integrals for BF-theory over the ring of integers of a totally imaginary field, which evaluate to natural arithmetic invariants associated to $\mathbb{G}_m$ and abelian varieties.
Comment: 5 pages
Comment: 5 pages
Externí odkaz:
http://arxiv.org/abs/1911.02236