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  "body": "\n## Summary\n\nThe 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 $q \\to 1$.\nNo field has one element, since a [[field|04C5ZBG1WT]] needs $0 \\ne 1$, so $\\F_1$ is not a field in the usual sense.\nMany counting formulas over $\\F_q$ become classical combinatorial formulas at $q = 1$, and $\\F_1$ is meant to explain this.\nSeveral 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.\n\n## Explanations\n\n### L1 human\n\nOver a finite field $\\F_q$, the number of lines through the origin in the plane $\\F_q^2$ is $q + 1$.\nAt $q = 1$ this gives $2$, the number of one-element subsets of a two-element set.\nMore 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 $\\binom{n}{k}$.\nSo a finite set behaves like a vector space over a field with one element, and the symmetric group $S_n$ plays the role of $GL_n(\\F_1)$.\nThere is no single agreed definition, and $\\F_1$ remains a guiding idea rather than one fixed object.\n",
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