代数数关于本原元的最小次数
Minimal Degrees of Algebraic Numbers with respect to Primitive Elements
摘要：
给定数域$L$，我们定义了代数数$v \in L$关于$L$的本原元的选择的次数.我们提出的问题，计算最小程度的$L$代数数，并检查这些值的程度$4$伽罗瓦扩展$\mathbb{Q}$和三次数域。证明了计算三次数域上非有理元的最小次数与求解经典丢番图问题（如全等数问题）以及理解椭圆曲线的各种算术性质密切相关.
英文摘要：
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.
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