The discriminant of a trinomial (Q2894092)

Let \(K\) be a field, \(P=X^n+ aX^k+b\in K[X]\) a trinomial and \(D\) the discriminant of \(P\). Assume that the characteristic of \(K\) is coprime to \(n(n-k)\). In the first part of the paper, using the well known formula \(D=(-1)^{n(n-1)/2}R(P', P)\), where \(R\) is the resultant and \(P'\) the derivative of \(P\), the authors give an explicit formula for \(D\) in terms of \(n\), \(k\), \(a\) and \(b\); this is the main result of the article.NEWLINENEWLINEIn the second part, supposing \(P\) irreducible, they present a second method to compute \(D\) based on the well known formula \(D=(-1)^{n(n-1)/2}N_{K(\alpha)/K}\big(P'(\alpha)\big)\), where \(\alpha\) is a root of \(P\) in an algebraic closure of \(K\) and \(N_{K(\alpha)/K}\) is the norm map in the separable extension \(K(\alpha)/K\), and a result in [\textit{G. R. Greenfield} and \textit{D. Drucker}, ``On the discriminant of a trinomial'', Linear Algebra Appl. 62, 105--112 (1984; Zbl 0552.12013)] that evaluates determinants of special matrices. As an application, the authors compute the (well known) discriminant of the \(m\)th cyclotomic field over \(\mathbb{Q}\). We note that analogous formulae of \(D\) are in the above paper of Greenfield and Drucker, and in [\textit{R. G. Swan}, ``Factorization of polynomials over finite fields'', Pac. J. Math. 12, 1099--1106 (1962; Zbl 0113.01701)].