Discriminant and integral basis of quintic fields defined by x5 + ax + b

Abstract

Let [Formula: see text] be an algebraic number field with [Formula: see text] a root of an irreducible trinomial [Formula: see text] belonging to [Formula: see text]. In this paper, we compute the highest power of each prime [Formula: see text] dividing the discriminant of [Formula: see text] in terms of powers of [Formula: see text] dividing [Formula: see text] and the discriminant of [Formula: see text] besides explicitly constructing a [Formula: see text]-integral basis of [Formula: see text]. These [Formula: see text]-integral bases lead to the construction of an integral basis of [Formula: see text] which is illustrated with examples.