Minimal Degrees of Algebraic Numbers with respect to Primitive Elements

Abstract

Given a number field $L$, we define the degree of an algebraic number $v\in L$ 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 $\mathbb{Q}$ 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.

arXiv commentAccepted to International Journal of Number Theory