论文标题
相对二次扩展的经典方法
A classical approach to relative quadratic extensions
论文作者
论文摘要
我们表明，我们可以从头开始开发，并且仅使用经典语言一种给定数字字段$ k $的相对二次扩展理论，这与众所周知的案例一样明确，容易，即$ k $是理性数字的领域。作为一个应用程序，我们证明了一项互惠定律，该法律表达给定二次方程式模型的解决方案的数量是不可或缺的理想$ \ mathfrak {a} $ $ k $的$ k $，其$ \ mathfrak {a} $ modulo是该方程式的歧视。我们研究了与相对二次扩展相关的各种$ L $函数。特别是，我们定义完全负代数的整数$Δ$的完全实数字段$ k $，即正方形模型〜$ 4 $，数字$ h（δ，k）$，它们具有经典的Hurwitz类数字的重要属性。在附录中，我们在高级单位局部数字字段上提供了希尔伯特符号的某些更深层属性的快速基本证明。
We show that we can develop from scratch and using only classical language a theory of relative quadratic extensions of a given number field $K$ which is as explicit and easy as for the well-known case that $K$ is the field of rational numbers. As an application we prove a reciprocity law which expresses the number of solutions of a given quadratic equation modulo an integral ideal $\mathfrak{a}$ of $K$ in terms of $\mathfrak{a}$ modulo the discriminant of the equation. We study various $L$-functions associated to relative quadratic extensions. In particular, we define, for totally negative algebraic integers $Δ$ of a totally real number field $K$ which are squares modulo~$4$, numbers $H(Δ,K)$, which share important properties of classical Hurwitz class numbers. In an appendix we give a quick elementary proof of certain deeper properties of the Hilbert symbol on higher unit groups of dyadic local number fields.
