  We show that the algebra and the endomotive of the quantum statistical
mechanical system of Bost--Connes naturally arises by extension of scalars from
the "field with one element" to rational numbers. The inductive structure of
the abelian part of the endomotive corresponds to the tower of finite
extensions of that "field", while the endomorphisms reflect the Frobenius
correspondences. This gives in particular an explicit model over the integers
for this endomotive, which is related to the original Hecke algebra
description. We study the reduction at a prime of the endomotive and of the
corresponding noncommutative crossed product algebra.
