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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== 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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== Expected properties ==

=== '''F'''₁ is not a field ===

'''F'''₁ cannot be a field because by definition all fields must contain two distinct elements, the [[additive identity]] zero and the [[multiplicative identity]] one. Even if this restriction is dropped (for instance by letting the additive and multiplicative identities be the same element), a ring with one element must be the [[zero ring]], which does not behave like a finite field. For instance, all [[Module (mathematics)|modules]] over the zero ring are isomorphic (as the only element of such a module is the zero element). However, one of the key motivations of '''F'''₁ is the description of sets as "'''F'''₁{{nbh}}vector spaces" – if finite sets were modules over the zero ring, then every finite set would be the same size, which is not the case. Moreover, the [[Spectrum of a ring|spectrum]] of the trivial ring is empty, but the spectrum of a field has one point.

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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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