Quadratic fields, Artin-Schreier extensions, and Bell numbers

In this article, we prove a modulo $p$ congruence which connects the class number of the quadratic field $\mathbb{Q}(\sqrt{(-1)^{(p-1)/2}p})$ and the trace of a certain element generating the Artin-Schreier extension of the field $\mathbb{F}_{p}$ 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 object is a purely combinatorial number counting the partitions of a finite set. Among such formulae, we prove a generalization of the so called "trace formula" which describes the special values of the Bell polynomials modulo $p$ by the trace mentioned above.