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The index of a monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ having a root $\theta$ is the index $[\mathbb{Z}_K:\mathbb{Z}[\theta]]$, where $\mathbb{Z}_K$ is the ring of algebraic integers of the number field $K=\mathbb{Q}(\theta)$. If $[\mathbb{Z}_K:\mathbb{Z}[\theta]]=1$, then $f(x)$ is monogenic. In this paper, we give necessary and sufficient conditions for a monic irreducible power compositional polynomial $f(x^k)$ belonging to $\mathbb{Z}[x]$, to be monogenic. As an application of our results, for a polynomial $f(x)=x^d+A\cdot h(x)\in\mathbb{Z}[x],$ with $d>1, \operatorname{deg} h(x)

Atomic effective pseudopotentials enable atomistic calculations at the level of accuracy of density functional theory for semiconductor nanostructures with up to fifty thousand atoms. Since they are directly derived from ab-initio calculations performed in the local density approximation (LDA), they inherit the typical underestimated band gaps and effective masses. We propose an empirical correction based on the modification of the non-local part of the pseudopotential and demonstrate good performance for bulk binary materials (InP, ZnS, HgTe, GaAs) and quantum dots (InP, CdSe, GaAs) with diameters ranging from 1.0 nm to 4.45 nm. Additionally, we provide a simple analytic expression to obtain accurate quasiparticle and optical band gaps for InP, CdSe, and GaAs QDs, from standard LDA calculation.

Suppose $m$ be a $12$-th power free integer. Let $K=\mathbb{Q}(\theta)$ be an algebraic number field defined by a complex root $\theta$ of an irreducible polynomial $x^{12}-m$ and $O_K$ be its ring of integers. In this paper, we determine the highest power of $p$ dividing the index of the subgroup $\Z[\theta]$ in $O_K$ and $p$-integral basis of $K$ for each prime $p$. These $p$-integral bases lead to the construction of an integral basis of $K$ which is illustrated with examples. In particular, when $m$ is a square free integer, we provide necessary and sufficient conditions for the set $\{1,\theta,\theta^2,\cdots,\theta^{10},\theta^{11}\}$ to be an integral basis of $K.$

Let $f(x)=x^n+ax^2+bx+c \in \Z[x]$ be an irreducible polynomial with $b^2=4ac$ and let $K=\Q(\theta)$ be an algebraic number field defined by a complex root $\theta$ of $f(x)$. Let $\Z_K$ deonote the ring of algebraic integers of $K$. The aim of this paper is to provide the necessary and sufficient conditions involving only $a,c$ and $n$ for a given prime $p$ to divide the index of the subgroup $\Z[\theta]$ in $\Z_K$. As a consequence, we provide families of monogenic algebraic number fields. Further, when $\Z_K \neq \Z[\theta]$, we determine explicitly the index $[\Z_K : \Z[\theta]]$ in some cases.

Let $f(x)=x^7+ax+b$ be an irreducible polynomial having integer coefficients and $K=\mathbb{Q}(\theta)$ be an algebraic number field generated by a root $\theta$ of $f(x)$. In the present paper, for every rational prime $p$, our objective is to determine the necessary and sufficient conditions involving only $a,~b$ so that $p$ is a divisor of the index of the field $K$. In particular, we provide sufficient conditions on $a$ and $b$, for which $K$ is non-monogenic. In a special case, we show that if either $8$ divides both $a\pm1$, $b$ or $32$ divides both $a+4$, $b$, then $K$ is non-monogenic. We illustrate our results through examples., Comment: arXiv admin note: text overlap with arXiv:2011.14348