Field with one element: Difference between revisions

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{{Short description|Theoretical object in mathematics}}

{{Use dmy dates|date=September 2020}}

In [[mathematics]], the '''field with one element''' is a suggestive name for an object that should behave similarly to a [[finite field]] with a single element, if such a field could exist. This object is denoted '''F'''₁, or, in a French–English pun, '''F'''_un.<ref>"[[wikt:un#French|un]]" is French for "one", and [[wikt:fun|fun]] is a playful English word. For examples of this notation, see, e.g. {{harvtxt|Le Bruyn|2009}}, or the links by Le Bruyn, Connes, and Consani.</ref> The name "field with one element" and the notation '''F'''₁ are only suggestive, as there is no field with one element in classical [[abstract algebra]]. Instead, '''F'''₁ refers to the idea that there should be a way to replace [[set (mathematics)|set]]s and [[Operation (mathematics)|operation]]s, the traditional building blocks for abstract algebra, with other, more flexible objects. Many theories of '''F'''₁ have been proposed, but it is not clear which, if any, of them give '''F'''₁ all the desired properties. While there is still no field with a single element in these theories, there is a field-like object whose [[characteristic (algebra)|characteristic]] is one.

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In 1957, Jacques Tits introduced the theory of [[building (mathematics)|buildings]], which relate [[algebraic group]]s to [[abstract simplicial complex]]es. One of the assumptions is a non-triviality condition: If the building is an ''n''{{nbh}}dimensional abstract simplicial complex, and if {{nowrap|''k'' < ''n''}}, then every ''k''{{nbh}}simplex of the building must be contained in at least three ''n''{{nbh}}simplices. This is analogous to the condition in classical [[projective geometry]] that a line must contain at least three points. However, there are [[Degeneracy (mathematics)|degenerate]] geometries that satisfy all the conditions to be a projective geometry except that the lines admit only two points. The analogous objects in the theory of buildings are called apartments. Apartments play such a constituent role in the theory of buildings that Tits conjectured the existence of a theory of projective geometry in which the degenerate geometries would have equal standing with the classical ones. This geometry would take place, he said, over a ''field of characteristic one''.<ref>{{harvtxt|Tits|1957}}.</ref> Using this analogy it was possible to describe some of the elementary properties of '''F'''₁, but it was not possible to construct it.

After Tits' initial observations, little progress was made until the early 1990s. In the late 1980s, [[Alexander Smirnov (mathematician)|Alexander Smirnov]] gave a series of talks in which he conjectured that the Riemann hypothesis could be proven by considering the integers as a curve over a field with one element. By 1991, Smirnov had taken some steps towards algebraic geometry over '''F'''₁,<ref name="Smirnov 1992">{{harvtxt|Smirnov|1992}}</ref> introducing extensions of '''F'''₁ and using them to handle the projective line '''P'''¹ over '''F'''₁.<ref name="Smirnov 1992"/> [[Algebraic number]]s were treated as maps to this '''P'''¹, and conjectural approximations to [[Riemann–Hurwitz formula|the Riemann–Hurwitz formula]] for these maps were suggested. These approximations imply solutions to important problems like [[abc conjecture|the abc conjecture]]. The extensions of '''F'''₁ later on were denoted as '''F'''_''q'' with {{nowrap|1=''q'' = 1^''n''}}. Together with [[Mikhail Kapranov]], Smirnov went on to explore how algebraic and [[number theory|number-theoretic]] constructions in prime characteristic might look in "characteristic one", culminating in an unpublished work released in 1995.<ref>{{harvtxt|Kapranov|Smirnov|1995}}</ref> In 1993, [[Yuri Manin]] gave a series of lectures on [[Riemann zeta function|zeta functions]] where he proposed developing a theory of algebraic geometry over '''F'''₁.<ref>{{harvtxt|Manin|1995}}.</ref> He suggested that zeta functions of [[algebraic variety|varieties]] over '''F'''₁ would have very simple descriptions, and he proposed a relation between the [[algebraic K-theory|K{{nbh}}theory]] of '''F'''₁ and the [[homotopy groups of spheres]]. This inspired several people to attempt to construct explicit theories of '''F'''₁{{nbh}}geometry.

The first published definition of a variety over '''F'''₁ came from [[Christophe Soulé]] in 1999,<ref name="Soule1999">{{harvtxt|Soulé|1999}}</ref> who constructed it using algebras over the [[complex number]]s and [[functor]]s from [[category (mathematics)|categories]] of certain rings.<ref name="Soule1999">{{harvtxt|Soulé|1999}}</ref> In 2000, Zhu proposed that '''F'''₁ was the same as '''F'''₂ except that the sum of one and one was one, not zero.<ref>{{harvtxt|Lescot|2009}}.</ref> Deitmar suggested that '''F'''₁ should be found by forgetting the additive structure of a ring and focusing on the multiplication.<ref>{{harvtxt|Deitmar|2005}}.</ref> Toën and Vaquié built on Hakim's theory of relative schemes and defined '''F'''₁ using [[symmetric monoidal category|symmetric monoidal categories]].<ref>{{harvtxt|Toën|Vaquié|2005}}.</ref> Their construction was later shown to be equivalent to Deitmar's by Vezzani.<ref>{{harvtxt|Vezzani|2010}}</ref> [[Nikolai Durov]] constructed '''F'''₁ as a commutative algebraic [[monad (category theory)|monad]].<ref>{{harvtxt|Durov|2008}}.</ref> Borger used [[descent (category theory)|descent]] to construct it from the finite fields and the integers.<ref>{{harvtxt|Borger|2009}}.</ref>

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== Motivations ==

=== Algebraic number theory ===

One motivation for '''F'''₁ comes from [[algebraic number theory]]. [[André Weil|Weil]]'s proof of the [[Riemann hypothesis for curves over finite fields]] starts with a curve ''C'' over a finite field ''k'', which comes equipped with a [[Function field of an algebraic variety|function field]] ''F'', which is a [[field extension]] of ''k''. Each such function field gives rise to a [[Hasse–Weil zeta function]] ''ζ''_''F'', and the Riemann hypothesis for finite fields determines the zeroes of ''ζ''_''F''. Weil's proof then uses various geometric properties of ''C'' to study ''ζ''_''F''.

The field of rational numbers '''Q''' is linked in a similar way to the [[Riemann zeta function]], but '''Q''' is not the function field of a variety. Instead, '''Q''' is the function field of the [[scheme (mathematics)|scheme]] {{nowrap|Spec '''Z'''}}. This is a one-dimensional scheme (also known as an [[algebraic curve]]), and so there should be some "base field" that this curve lies over, of which '''Q''' would be a [[field extension]] (in the same way that ''C'' is a curve over ''k'', and ''F'' is an extension of ''k''). The hope of '''F'''₁{{nbh}}geometry is that a suitable object '''F'''₁ could play the role of this base field, which would allow for a proof of the [[Riemann hypothesis]] by mimicking Weil's proof with '''F'''₁ in place of ''k''.

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* [http://cage.ugent.be/~kthas/Fun F_un Mathematics], Lieven le Bruyn, [[Thas, Koen|Koen Thas]].

* Vanderbilt conference on [http://www.math.vanderbilt.edu/~ncgoa/workshop2008.html Noncommutative Geometry and Geometry over the Field with One Element] {{Webarchive|url=https://web.archive.org/web/20131212171146/http://www.math.vanderbilt.edu/~ncgoa/workshop2008.html |date=12 December 2013 }} ([http://www.math.vanderbilt.edu/~ncgoa/schedule_workshop08.pdf Schedule] {{Webarchive|url=https://web.archive.org/web/20120215091922/http://www.math.vanderbilt.edu/~ncgoa/schedule_workshop08.pdf |date=15 February 2012 }})

* [httphttps://noncommutativegeometry.blogspot.com/2008/05/ncg-and-fun.html NCG and F_un], by [[Alain Connes]] and K. Consani: summary of talks and slides

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