---
tag: 9QP9ZAGH6T
kind: concept
name: Field with one element
msc: [14A23]
sources:
  - url: https://en.wikipedia.org/wiki/Field_with_one_element
same_as:
  wikidata: Q3654641
  wikipedia: 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|0ZXZ82FTX2]] $\F_q$ in the limit $q \to 1$.
No field has one element, since a [[field|04C5ZBG1WT]] 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.

## Explanations

### L1 human

Over a finite field $\F_q$, the number of lines through the origin in the plane $\F_q^2$ is $q + 1$.
At $q = 1$ this gives $2$, the number of one-element subsets of a two-element set.
More generally, the number of $k$-dimensional subspaces of $\F_q^n$ is a polynomial in $q$ whose value at $q = 1$ is the binomial coefficient $\binom{n}{k}$.
So a finite set behaves like a vector space over a field with one element, and the symmetric group $S_n$ plays the role of $GL_n(\F_1)$.
There is no single agreed definition, and $\F_1$ remains a guiding idea rather than one fixed object.