Minimal Degrees of Algebraic Numbers with respect to Primitive Elements

TL;DR

This paper introduces a new concept of minimal degrees of algebraic numbers relative to primitive elements in a number field, exploring their computation in specific Galois and triquadratic extensions and linking to classical Diophantine problems.

Contribution

It defines the minimal degree of algebraic numbers with respect to primitive elements and investigates their properties in degree 4 Galois and triquadratic fields, connecting to Diophantine equations.

Findings

Computing minimal degrees in triquadratic fields relates to the congruent number problem.

Minimal degrees in certain extensions reveal deep arithmetic properties.

The study links algebraic number theory with classical Diophantine problems.

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.