Field with one element: Difference between revisions

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{{Use dmy dates|date=July 2013}}

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 sets and operations, 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.

'''F'''₁ cannot be a field because all fields must contain two distinct elements, the [[additive identity]] zero and the [[multiplicative identity]] one. Even if this restriction is dropped, a ring with one element must be the [[zero ring]], which does not behave like a finite field. Instead, mostMost proposed theories of '''F'''₁ replace abstract algebra entirely. Mathematical objects such as [[vector space]]s and [[polynomial ring]]s can be carried over into these new theories by mimicking their abstract properties. This allows the development of [[commutative algebra]] and [[algebraic geometry]] on new foundations. One of the defining features of theories of '''F'''₁ is that these new foundations allow more objects than classical abstract algebra, one of which behaves like a field of characteristic one.

The possibility of studying the mathematics of '''F'''₁ was originally suggested in 1956 by [[Jacques Tits]], published in {{harvnb|Tits|1957}}, on the basis of an analogy between symmetries in [[projective geometry]] and the combinatorics of [[simplicial complex]]es. '''F'''₁ has been connected to [[noncommutative geometry]] and to a possible proof of the [[Riemann hypothesis]]. 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.

==History==

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==Motivations==

===Algebraic number theory===

One motivation for '''F'''₁ comes from [[algebraic number theory]]. Weil's proof of the [[Riemann hypothesis for curves over finite fields]] startedstarts 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]] {{math|ζ_''F''}}, and the Riemann hypothesis for finite fields determines the zeroes of {{math|ζ_''F''}}. Weil's proof then useduses varius geoemtricgeometric properties of ''C'' to study {{math|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]] {{math|Spec '''Z'''}}. This is [[Krulla dimension|one-dimensional scheme (a.k.a. an [[algebraic curve]]), and so there should be some "base field" which 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''). However, every field contains '''Q''' as a subfield, so no such field exists. The hope of '''F'''₁-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''.

===Arakelov geometry===

Geometry over a field with one element is also motivated by [[Arakelov geometry]], where [[Diophantine equations]] are studied using tools from [[complex geometry]]. The theory involves complicated comparisons between finite fields and the complex numbers. Here the existence of '''F'''₁ is useful for technical reasons.

==Expected Properties==

==='''F'''₁ is expectednot toa have the following properties.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, every [[Module (mathematics)|module]] over the zero ring is 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-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.

===Other properties===

* [[Finite set]]s are both [[affine space]]s and [[projective space]]s over '''F'''₁.

* [[Pointed set]]s are [[vector space]]s over '''F'''₁.<ref>[http://sbseminar.wordpress.com/2007/08/14/the-field-with-one-element Noah Snyder, The field with one element, Secret Blogging Seminar, 14 August 2007.]</ref>