On monogenity of number fields and Galois group
Anuj Jakhar, Ravi Kalwaniya, Srinivas Kotyada · International Journal of Number Theory · 2025
Let [Formula: see text] denote the ring of algebraic integers of an algebraic number field [Formula: see text], where [Formula: see text] is a root of an irreducible polynomial [Formula: see text], [Formula: see text]. We say [Formula: see text] is monogenic if [Formula: see text] is a basis for [Formula: see text]. In this paper, we explicitly compute the discriminant of [Formula: see text] and give necessary and sufficient conditions involving only [Formula: see text] for [Formula: see text] to be monogenic. Moreover, we characterize all the primes dividing the index of the subgroup [Formula: see text] in [Formula: see text]. As an application, we also provide a class of monogenic polynomials having nonsquare-free discriminant and Galois group [Formula: see text], the symmetric group on n letters. In the end, we state a conjecture and show its implications.