Minimal Degrees of Algebraic Numbers with respect to Primitive Elements

Given a number field L, we define the degree of an algebraic number v ∈ L with respect to a choice of a primitive element of L. We propose the question of computing the minimal degrees of algebraic numbers in L, and examine these values in degree 4 Galois extensions over ℚ 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.