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 monomial in a root $\theta$ of the Artin-Schreier polynomial $\theta^{p}-\theta-1$ over 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 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.