AI 中文总结
本文基于数域整数环的高阶代数K-群结构，引入K-等价与几乎K-等价，结合Rost-Voevodsky定理等，在p≠2时改进Komatsu的结果，明确相关等价条件及算术信息的共鸣关系。
AI 中文摘要
本文基于数域整数环的高阶代数K-群的结构，引入数域之间的新等价关系，称为K-等价与几乎K-等价，并研究它们与算术等价、局部整等价的关系。在R. Perlis研究的算术等价数域的经典性质及Komatsu的开创性工作基础上，我们将Rost-Voevodsky定理（Quillen-Lichtenbaum猜想）应用于奇素数p，从更现代的角度在连续平展上同调的框架下分析代数K-群。作为主要结果，通过利用整体伽罗瓦群的置换表示和局部分解群的数据，我们在p≠2的范围内改进了Komatsu之前的结果，并描述了数域为（几乎）K-等价的条件。通过这些进展，我们阐明了高阶K-群中反映的算术信息如何与ζ函数的特殊值及伽罗瓦表示产生共鸣。
英文摘要
In this paper, based on the structure of higher algebraic $K$-groups of the rings of integers of number fields, we introduce new equivalence relations between number fields called $K$-equivalence and almost $K$-equivalence, and investigate their relationships with arithmetical equivalence and local integral equivalence. Building upon the classical properties of arithmetically equivalent number fields studied by R. Perlis and on the pioneering work by Komatsu, we apply the Rost-Voevodsky theorem (the Quillen-Lichtenbaum conjecture) for odd primes $p$, thereby analyzing algebraic $K$-groups within the framework of continuous étale cohomology from a more modern perspective. As our main results, by utilizing permutation representations of global Galois groups and the data of local decomposition groups, we refine Komatsu's previous results in the range $p \neq 2$ and describe the conditions for number fields to be (almost) $K$-equivalent. Through these developments, we clarify how the arithmetic information reflected in higher $K$-groups resonates with the special values of zeta functions and Galois representations.
Comments20 pages
