2019/12/10 by Yoshinosuke Hirakawa, Hirakawa, Yoshinosuke
Mathematics · #11M38 #11R29 #12E20 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #primary 11B73 #secondary 11B83
paper · pdf · doi:10.48550/arxiv.1912.04647
openalex publication_date 2019/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we prove a modulo p congruence which connects the class number of the quadratic field ℚ(√(-1)(p-1)/2p) and the trace of a certain monomial in a root θ of the Artin-Schreier polynomial θp-θ-1 over the field \mathbbFp of p elements. This formula has a flavor of Dirichlet's class number formula which connects the class number and the L-value. The proof of our formula is based on several formulae satisfied by the Bell number, where the latter is defined as the number of partitions of \ 1, 2, ..., n \ and a purely combinatorial object. Among such formulae, we prove a generalization of the so called ``trace formula'' due to Barsky and Benzaghou which describes the special values of the Bell polynomials modulo p by the trace mentioned above.