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.

中文翻译：

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关于原始元素的代数数的最小度数

给定一个数字字段大号, 我们定义代数数的次数v ∈ 大号关于一个原始元素的选择大号. 我们提出了计算代数数的最小度数的问题大号, 并检查这些值的度数4伽罗瓦扩展ℚ和三二次数字段。我们表明，计算三二次数域中非有理元素的最小度数与解决经典丢番图问题（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。