arXiv subjects
Publications and source records attributed to Mohammed Seddik.
In this paper, we investigate the common index divisors of cyclic cubic fields. Let $a,b,c,d$ and $k$ are integers, we then solve the following Thue cubic equations:: \[ax^3+bx^2y+cxy^2+dy^3= k\ \] when $a,bc+d$ are odd and $3$ doesn't divide $v_2(k)$.
Let $\mathbb{K}$ be a number field of degree $n$ over $\mathbb{Q}$. Let $\widehat{\mathbb{A}}$ be the set of integers of $\mathbb{K}$ which are primitive over $\mathbb{Q}$ and $I(\mathbb{K})$ be its index. Gunji and McQuillan defined the following integer $i(\mathbb{K})=\underset{θ\in \widehat{\mathbb{A}}}{\text{lcm}}\;i(θ)$, where $i(θ)=\underset{x\in \mathbb{Z}}{\text{gcd}}\;F_θ(x)$ and $F_θ(x)$ is the characteristic polynomial of $θ$ over $\mathbb{Q}$. We prove that if $p$ is a prime number less than or equal to $n$ then there exists a number field $\mathbb{K}$ of degree $n$ for which $p$ divides $i(\mathbb{K})$. We compute $i(\mathbb{K})$ for cubic fields. Also we determine $I(\mathbb{K})$ and $i(\mathbb{K})$ for families of simplest number fields of degree less than $7$. We give also answers to questions one and two in \cite{Kihel}. Furthermore, we give a counter example to Theorem 11 in \cite{Kihel} and we discuss their conjecture.
We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. In this paper, we study the common index divisor $I(\mathbb K_m)$ and determine explicitly the prime ideal decomposition for any prime number in any simplest quartic number fields $\mathbb{K}_m$. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant $\leq x$ and given index.