The index of a quartic field defined by a trinomial X-4 + aX plus b

摘要

Consider an irreducible quartic polynomial of the form X-4 + aX + b, where a, b is an element of Z satisfy v(p) (a) 3 or v(p) (b) 4, where v(p) denotes the exact power of a rational prime p that divides an integer. Such a polynomial is called a p-minimal quartic. Let K be a field defined by a p-minimal quartic. In this paper, we use p-integral bases and introduce the concept of p-index forms in order to determine the field index of K via the coefficients of a defining polynomial in terms of certain congruence conditions.