Field with one element

Summary

The field with one element, written $\F_1$, is a hoped-for object that would behave like a finite field $\F_q$ in the limit $q \to 1$. No field has one element, since a field needs $0 \ne 1$, so $\F_1$ is not a field in the usual sense. Many counting formulas over $\F_q$ become classical combinatorial formulas at $q = 1$, and $\F_1$ is meant to explain this. Several competing theories of geometry over $\F_1$ exist, partly motivated by the hope of treating $\operatorname{Spec} \Z$ like a curve over a finite field.