标题： 指数同余符号 显示英文标题

标题： The Exponential Congruence Symbol

摘要： 在本工作中，我们研究广义的k次幂符号(a/n)_k，并提供了其代数性质的全面集合。 结果根据其对三个主要参数a、n和k的依赖性进行分类。 特别是，我们讨论了关于参数a的可乘性、倒数、幂兼容性以及模n下的不变性。 对于n，我们研究了分解性质、在素数幂上的行为、正交关系以及Kummer分裂准则。 关于k，我们包括经典符号的特例、k次互反律、阶之间的关系以及与单位根的嵌入。 此外，我们通过提供新的关键结果扩展了现有理论，包括在特征上的加法行为、莫比乌斯过滤、与Carmichael函数和欧拉函数的兼容性以及与狄利克雷L级数的联系。 最后，我们分析了a、n和k均为素数的情况，并提出了推广经典互反律、弗罗贝尼乌斯自同构和佐藤-塔特定律的混合结果。 这些结果统一并扩展了对k次幂符号的先前研究，并为进一步的算术、代数和解析研究奠定了基础。 显示英文摘要

摘要： In this work, we study the generalized k-th power symbol (a/n)_k and present a comprehensive collection of its algebraic properties. The results are classified according to their dependence on the three main parameters a, n, and k. In particular, we discuss multiplicativity, inversion, power compatibility, and invariance modulo n with respect to the parameter a. For n, we examine factorization properties, behavior on prime powers, orthogonality relations, and Kummer splitting criteria. Regarding k, we include specialization to classical symbols, k-th reciprocity laws, relations between orders, and embedding into roots of unity. Moreover, we extend the existing theory by providing new essential results, including additive behavior under characters, Mobius filtering, compatibility with Carmichael and Euler functions, and connections with Dirichlet L-series. Finally, we analyze the case where a, n, and k are primes and present mixed results that generalize classical reciprocity laws, Frobenius automorphisms, and Sato-Tate distributions. These results unify and extend previous studies on k-th power symbols and offer a foundation for further arithmetic, algebraic, and analytic investigations.