The index of a quartic field defined by a trinomial X4 + aX + b

Abstract

Consider an irreducible quartic polynomial of the form [Formula: see text], where [Formula: see text] satisfy [Formula: see text] or [Formula: see text], where [Formula: see text] denotes the exact power of a rational prime [Formula: see text] that divides an integer. Such a polynomial is called a [Formula: see text]-minimal quartic. Let [Formula: see text] be a field defined by a [Formula: see text]-minimal quartic. In this paper, we use [Formula: see text]-integral bases and introduce the concept of [Formula: see text]-index forms in order to determine the field index of [Formula: see text] via the coefficients of a defining polynomial in terms of certain congruence conditions.