On the discriminant of pure number fields

Abstract

Let K = Q( n √ a) be an extension of degree n of the field Q of rational numbers, where the integer a is such that for each prime p dividing n either p ∤ a or the highest power of p dividing a is coprime to p; this condition is clearly satisfied when a, n are coprime or a is squarefree. The paper contains an explicit formula for the discriminant of K involving only the prime powers dividing a, n.

Discriminant and Integral basis of sextic fields defined by x^6+ax+b

“…In 2020, Jakhar et al gave an explicit construction of an integral basis of all those n-th degree pure fields Q( n √ a) which are such that for each prime p dividing n, either p a or p does not divide v p (a), where v p (a) stands for the highest power of p dividing a; clearly this condition is satisfied when either a, n are coprime or a is squarefree (cf. [12], [13]). A different approach using p-integral basis defined below has been followed by A. Alaca, S. Alaca and K. S. Williams in [1], [2], [3] to construct integral bases of all cubic fields and all those quartic as well as quintic fields which are generated over Q by a root of an irreducible trinomial of the type x n + ax + b belonging to Z[x] with n = 4, 5.…”

“…12 × 8539. By case E18 of TableI, {1, θ, θ 2 , θ 3 /2, θ 4 /2, θ 5 /2} is a 2-integral basis of K. Since 8539 is a prime, it does not divide ind θ.…”