Eulerian digraphs and the Schönemann-Gauss theorem

In 19th century, Fermat's little theorem ``$a^p\equiv a({\rm mod}\;p)$ for $a\in\mathbb Z$, $p$ prime'' was generalized in two directions: Schonemann proved a corresponding congruence for the coefficients of monic polynomials, whereas Gauss found a congruence result with $p$ replaced by any $n\in\mathbb N$. Here, we shall give an elementary proof of the common generalization of these two results.