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Monogenity and Power Integral Bases: Recent Developments [PDF]

Axioms
Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area.
István Gaál
doaj   +4 more sources

On relative pure cyclic fields with power integral bases [PDF]

Mathematica Bohemica, 2023
Let $L = K(\alpha)$ be an extension of a number field $K$, where $\alpha$ satisfies the monic irreducible polynomial $P(X)=X^p-\beta$ of prime degree belonging to $\mathfrak{o}_K[X]$ ($\mathfrak{o}_K$ is the ring of integers of $K$).
Mohammed Sahmoudi, Mohamed E. Charkani
doaj   +6 more sources

On monogenity of certain pure number fields of degrees $2^r\cdot3^k\cdot7^s$ [PDF]

Mathematica Bohemica, 2023
Let $K = \mathbb{Q} (\alpha) $ be a pure number field generated by a complex root $\alpha$ of a monic irreducible polynomial $ F(x) = x^{2^r\cdot3^k\cdot7^s} -m \in\bb{Z}[x]$, where $r$, $k$, $s$ are three positive natural integers.
Hamid Ben Yakkou, Jalal Didi
doaj   +2 more sources

Investigating Monogenity in a Family of Cyclic Sextic Fields

Mathematics
Jones characterized, among others, monogenity of a family of cyclic sextic polynomials. Our purpose is to study monogenity of the family of corresponding sextic number fields.
István Gaál
doaj   +2 more sources

LOCAL EXTENSIONS WITH IMPERFECT RESIDUE FIELD [PDF]

Ural Mathematical Journal, 2019
The paper deals with some aspects of general local fields and tries to elucidate some obscure facts. Indeed, several questions remain open, in this domain of research, and literature is getting scarce.
Akram Lbekkouri
doaj   +3 more sources

On common index divisors and monogenity of septic number fields defined by trinomials of type $x^7+ax^5+b$ [PDF]

Mathematica Bohemica
Let $K $ be a septic number field generated by a root $\theta$ of an irreducible polynomial $ F(x)= x^7+ax^5+b \in\mathbb Z[x]$. In this paper, we explicitly characterize the index $i(K)$ of $K$.
Hamid Ben Yakkou
doaj   +3 more sources

On the Monogenity of Quartic Number Fields Defined by x⁴ + ax² + b

Mathematics
For any quartic number field K generated by a root α of an irreducible trinomial of type x4+ax2+b∈Z[x], we characterize when Z[α] is integrally closed. Also for p=2, 3, we explicitly give the highest power of p dividing i(K), the common index divisor of ...
Lhoussain El Fadil, István Gaál
doaj   +2 more sources

On the index divisors and monogenity of certain nonic number fields

Rocky Mountain Journal of Mathematics, 2023
In this paper, for any nonic number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^9+ax+b \in \mathbb{Z}[x]$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$.
Kchit, Omar
core   +2 more sources

A Study of monogenity of Binomial Composition

Acta Arithmetica
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 ...
Jakhar, Anuj, Kalwaniya, Ravi, Yadav, Prabhakar   +2 more
core   +2 more sources

On indices and monogenity of quartic number fields defined by quadrinomials

Mathematica Bohemica
Consider a quartic number field $K$ generated by a root of an irreducible quadrinomial of the form $ F(x)= x^4+ax^3+bx+c \in \Z[x]$. Let $i(K)$ denote the index of $K$. Engstrom \cite{Engstrom} established that $i(K)=2^u \cdot 3^v$ with $u \le 2$ and $v \
Yakkou, Hamid Ben
core   +2 more sources