Let $K = \mathbb{Q} (α) $ be a pure number field generated by a complex root $α$ a monic irreducible polynomial $ F(x) = x^{p^r} -m$, with $ m \neq 1 $ is a square free rational integer, $p$ is a rational prime integer, and $r$ is a positive integer. In this paper, we study the monogenity of $K$. We prove that if {$ν_p(m^p-m)=1$}, then $K$ is monogenic. But if $r\ge p$ and {$ν_p(m^{p}-m)> p$}, then $K$ is not monogenic. Some illustrating examples are given.
Let $m$ be a square-free integer ($m\neq 0,\pm 1$). We show that the structure of the integral bases of the fields $K=Q(\sqrt[n]{m})$ are periodic in $m$. For $3\leq n\leq 9$ we show that the period length is $n^2$. We explicitly describe the integral bases, and for $n=3,4,5,6,8$ we explicitly calculate the index forms of $K$. This enables us in many cases to characterize the monogenity of these fields. Using the explicit form of the index forms yields a new technic that enables us to derive new results on monogenity and to get several former results as easy consequences. For $n=4,6,8$ we give…
We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly describe an integral basis of the composite fields. We construct the index form, describe their factors and prove that the monogenity of the composite fields imply certain divisibility conditions on the parameters involved. These conditions usually can not hold, which implies the non-monogenity of the fields. The fields that we consider are higher degree number fields, o…
Let $K=\Q(θ)$ be a number field generated by a complex root $þ$ of a monic irreducible trinomial $F(x) = x^n+ax+b \in \Z[x]$. There is an extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by trinomials. Jhorar and Khanduja studied the integral closedness of $\Z[þ]$. But if $ \Z[þ]$ is not integrally closed, then Jhorar and Khanduja's results cannot answer on the monogenity of $K$. In this paper, based on Newton polygon techniques, we deal with the problem of monogenity of $K$. More precisely, when $\Z_K …
For a number field $K$ defined by a trinomail $F(x) = x^7+ax+b \in \mathbb{Z}[x]$, Jakhar and Kumar gave some necessary conditions on $a$ and $b$, which guarantee the non-monogenity of $K$ \cite{A6}. In this paper, for every prime integer $p$, we characterize when $p$ is a common index divisor of $K$. In particular, if any one of these conditions holds, then $K$ is not monogenic. In such a way our proposed results extend those of Jakhar and Kumar.
In this paper, we deal with the problem of monogenity of number fields defined by monic irreducible trinomials $F(x)=x^{12}+ax^m+b\in \mathbb{Z}[x]$ with $1\leq m\leq11$. We give sufficient conditions on $a$, $b$, and $m$ so that the number field $K$ is not monogenic. In particular, for $m=1$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$ and we provide a partial answer to the Problem $22$ of Narkiewicz \cite{Nar} for these number fields. Our results are illustrated by computational examples.
Let $K = \mathbb{Q} (α) $ be a pure number field generated by a root $α$ of a monic irreducible polynomial $ F(x) = x^{2^u\cdot 3^v\cdot 5^t}-m$, with $ m \neq \pm 1 $ a square free rational integer, $u$, $v$ and $t$ three positive integers. In this paper, we study the monogenity of $K$. We prove that if $m\not\equiv 1\md4$, $m\not\equiv \pm 1\md9$, and $m\not\in\{\pm 1, \pm 7\}\md{25}$, then $K$ is monogenic. But if {$m\equiv 1\md{4}$} or $m\equiv 1\md9$ or $m\equiv -1\md9$ and $u=2k$ for some odd integer $k$ or $u\ge 2$ and $m\equiv 1\md{25}$ or $m\equiv -1\md{25}$ and $u=2k$ for some odd in…
The main goal of this paper is to provide a complete answer to the Problem 22 of Narkiewicz \cite{Nar} for any sextic number field $K$ generated by a complex root $α$ of a monic irreducible trinomial $F(x) = x^6+ax^5+b \in \mathbb{Z}[x]$. Namely we calculate the index of the field $K$. In particular, if $i(K)\neq 1$, then $K$ is not mongenic. Finally, we illustrate our results by some computational examples.
Let $K$ be a number field generated by a root $þ$ of a monic irreducible trinomial $F(x) = x^n+ax^{m}+b \in \Z[x]$. In this paper, we study the problem of $K$. More precisely, we provide some explicit conditions on $a$, $b$, $n$, and $m$ for which $K$ is not monogenic. As applications, we show that there are infinite families of non-monogenic number fields defined by trinomials of degree $n=2^r\cdot3^k$ with $r$ and $k$ two positive integers. We also give infinite families of non-monogenic sextic number fields defined by trinomials. Some illustrating examples are giving at the end of this pape…
Let $K = \mathbb{Q} (α) $ be a pure number field generated by a complex root $α$ of a monic irreducible polynomial $ F(x) = x^{2^u\cdot 3^v}-m$, with $m \neq \pm 1$ a square free rational integer, $u$, and $v$ two positive integers. In this paper, we study the monogenity of $K$. The case $u=0$ or $v=0$ has been previously studied. We prove that if $m\not\equiv 1$ (mod4) and $m\not\equiv \pm 1$ (mod9), then $K$ is monogenic. But if $m\equiv 1$ (mod4) or $m\equiv 1$ (mod9), then $K$ is not monogenic. Some illustrating examples are given too.
Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{60}-m\in \mathbb{Z}[x]$, with $m\neq \pm1$ a square free integer. In this paper, we study the monogeneity of $K$. We prove that if $m\not\equiv 1\md{4}$, $m\not\equiv \mp 1 \md{9} $ and $\overline{m}\not\in\{\mp 1,\mp 7\} \md{25}$, then $K$ is monogenic. But if $m\equiv 1\md{4}$, $m\equiv \mp1 \md{9}$, or $m\equiv \mp 1\md{25}$, then $K$ is not monogenic. Our results are illustrated by examples.
Reduced ideals have been defined in the context of integer rings in quadratic number fields, and they are closely tied to the continued fraction algorithm. The notion of this type of ideal extends naturally to number fields of higher degree. In the case of pure cubic fields, generated by cube roots of integers, a convenient integral basis provides a means for identifying reduced ideals in these fields. We define integer sequences whose terms are in correspondence with some of these ideals, suggesting a generalization of continued fractions.
A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields. As an application we show that for at least two arithmetic directions every elliptic curve possesses a non-zero arithmetic PDE Manin map of order 1; such maps do not exist in the arithmetic ODE case. Similarly we construct and study "genuinely PDE" differential modular forms. As further applications we derive a Theorem of the Kernel and a Reciprocity Theore…
