Alladi Ramakrishnan Hall

On discriminant, integral basis and common index divisors of certain quintic number fields

Sumandeep Kaur

Panjab University

Let $K=\mathbb{Q}(\theta)$ be an algebraic number field with $\theta$ a root of an irreducible quintic polynomial of the type $x^5+ax+b\in\mathbb{Z}[x]$. Let $A_K$ stand for the ring of algebraic integers of $K$. If ind $ \theta$ denotes the index of the subgroup $\mathbb{Z}[\theta]$ in $A_K $ and $i(K)$ stands for the index of the field $K$ defined by
$$ i(K) = \gcd\{\text {ind} ~\alpha \mid { K=\mathbb{Q}(\alpha),~ \alpha\in A_K} \}.$$

Then a prime number $p$ dividing $i(K)$ is called a prime common index divisor of $K$. In this talk, for every rational prime $p$, we provide necessary and sufficient conditions on $a,~b$ so that $p$ is a common index divisor of $K$. In particular, we give sufficient conditions on $a,~b$ for which $K$ is non-monogenic. Also, we compute the highest power of each prime $p$ dividing the discriminant of $K$ besides explicitly constructing an integral basis of $K$. This talk is based on joint work with A. Jakhar, S. K. Khanduja and S. Kumar.