Minimal degrees of algebraic numbers with respect to primitive elements

Abstract

Given a number field [Formula: see text], we define the degree of an algebraic number [Formula: see text] with respect to a choice of a primitive element of [Formula: see text]. We propose the question of computing the minimal degrees of algebraic numbers in [Formula: see text], and examine these values in degree [Formula: see text] Galois extensions over [Formula: see text] and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves.