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{{Short description|Theoretical object in mathematics}}

{{Use dmy dates|date=JulySeptember 20132020}}

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[[set (mathematics)|set]]s and operations[[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.

'''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 does, 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 ==

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 whichthat 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.

A separate inspiration for '''F'''₁ came from [[algebraic number theory]]. Weil's proof of the [[Riemann hypothesis for curves over finite fields]] started with a curve ''C'' over a finite field ''k'', took its product {{nowrap|''C'' ×_''k'' ''C''}}, and then examined its diagonal. If the integers were a curve over a field, the same proof would prove the [[Riemann hypothesis]]. The integers '''Z''' are [[Krull dimension|one-dimensional]], which suggests that they may be a curve, but they are not an algebra over any field. One of the conjectured properties of '''F'''₁ is that '''Z''' should be an '''F'''₁-algebra. This would make it possible to construct the product {{nowrap|'''Z''' ×_'''F'''₁ '''Z'''}}, and it is hoped that the Riemann hypothesis for '''Z''' can be proved in the same way as the Riemann hypothesis for a curve over a finite field.

InThe 1993,first [[Yuripublished Manin]] gave a seriesdefinition of lectures on [[Riemann zeta function|zeta functions]] where he proposed developing a theory of algebraic geometryvariety over '''F'''₁. came from [[Christophe Soulé]] in 1999,<ref name="Soule1999">{{harvtxt|ManinSoulé|19951999}}.</ref> Hewho suggestedconstructed thatit zetausing functions of varietiesalgebras over '''F'''₁ would have very simple descriptions, and he proposed a relation between the [[algebraiccomplex K-theory|K-theorynumber]] of '''F'''₁s and the [[homotopyfunctor]]s groupsfrom of[[category spheres(mathematics)|categories]]. Thisof inspiredcertain severalrings.<ref people to attempt to construct '''F'''<subname="Soule1999">1{{harvtxt|Soulé|1999}}</subref>. 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> Soulé constructed it using algebras over the complex numbers and functors from categories of certain rings.<ref name="Soule1999">{{harvtxt|Soulé|1999}}</ref> Borger used [[descent (category theory)|descent]] to construct it from the finite fields and the integers.<ref>{{harvtxt|Borger|2009}}.</ref>

Another angle comes from [[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.

[[Alain Connes]] and [[Caterina Consani]] developed both Soulé and Deitmar's notions by "gluing" the category of multiplicative monoids[[monoid]]s and the category of rings to create a new category <math>\mathfrak{M}\mathfrak{R},</math> then defining '''F'''₁-{{nbh}}schemes to be a particular kind of representable functor on <math>\mathfrak{M}\mathfrak{R}.</math><ref>{{harvtxt|Connes|Consani|2010}}.</ref> Using this, they managed to provide a notion of several number-theoretic constructions over '''F'''₁ such as motives and field extensions, as well as constructing [[Groups of Lie type#Chevalley groups|Chevalley groups]] over '''F'''_1². Along with [[Matilde Marcolli]], Connes- and Consani have also connected '''F'''₁ with [[noncommutative geometry]].<ref>{{harvtxt|Connes|Consani|Marcolli|2009}}</ref> It has also been suggested to have connections to the [[unique games conjecture]] in [[computational complexity theory]].<ref>{{citation|url=https://gilkalai.wordpress.com/2018/01/10/subhash-khot-dor-minzer-and-muli-safra-proved-the-2-to-2-games-conjecture/|title=Subhash Khot, Dor Minzer and Muli Safra proved the 2-to-2 Games Conjecture|work=Combinatorics and more|first=Gil|last=Kalai|authorlink=Gil Kalai|date=2018-01-10}}</ref>

By 1991, Alexander Smirnov had taken some steps towards algebraic geometry over '''F'''₁.<ref>{{harvtxt|Smirnov|1992}}</ref> He introduced extensions of '''F'''₁ and used them to handle the projective line '''P'''¹ over '''F'''₁. [[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 very profound assertions like [[abc conjecture|the abc conjecture]]. The extensions of '''F'''₁ later on were denoted<ref>{{harvtxt|Kapranov|Smirnov|1995}}</ref> as '''F'''_''q'' with ''q'' = 1^''n''.

Oliver Lorscheid, along with others, has recently achieved TitTits's original aim of describing Chevalley groups over '''F'''₁ by introducing objects called blueprints, which are a simultaneous generalisation of both [[semiring]]s and monoids.<ref>{{harvtxt|Lorscheid|2018a}}< name=":0"/ref><ref>{{harv|Lorscheid|2018b}}</ref> These are used to define so-called "blue schemes", one of which is Spec '''F'''₁.<ref>{{harvtxt|Lorscheid|2016}}</ref> Lorscheid's ideas depart somewhat from other ideas of groups over '''F'''₁, in that the '''F'''₁-{{nbh}}scheme is not itself the Weyl group of its base extension to normal schemes. Lorscheid first defines the Tits category, a full subcategory of the category of blue schemes, and defines the "Weyl extension" , a functor from the Tits category to '''Set'''. A Tits-WeylTits–Weyl model of an algebraic group <math>\mathcal{G}</math> is a blue scheme ''G'' with a group operation whichthat is a morphism in the Tits category, whose base extension is <math>\mathcal{G}</math> and whose Weyl extension is isomorphic to the Weyl group of <math>\mathcal{G}.</math>

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 varieties over '''F'''₁ would have very simple descriptions, and he proposed a relation between the [[algebraic K-theory|K-theory]] of '''F'''₁ and the [[homotopy groups of spheres]]. This inspired several people to attempt to construct '''F'''₁. 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> Soulé constructed it using algebras over the complex numbers and functors from categories of certain rings.<ref name="Soule1999">{{harvtxt|Soulé|1999}}</ref> Borger used [[descent (category theory)|descent]] to construct it from the finite fields and the integers.<ref>{{harvtxt|Borger|2009}}.</ref>

'''F'''₁-{{nbh}}geometry has been linked to [[tropical geometry]], via the fact that semirings (in particular, tropical semirings) arise as quotients of some monoid semiring '''N'''[''A''] of finite formal sums of elements of a monoid ''A'', which is itself an '''F'''₁-{{nbh}}algebra. This connection is made explicit by Lorscheid's use of blueprints.<ref>{{harvtxt|Lorscheid|2015}}</ref> The Giansiracusa brothers have constructed a tropical scheme theory, for which their category of tropical schemes is equivalent to the category of Toën-VaquiéToën–Vaquié '''F'''₁-{{nbh}}schemes.<ref>{{harvtxt|Giansiracusa|Giansiracusa|2016}}</ref> This category embeds [[faithful functor|faithfully]], but not [[full functor|fully]], into the category of blue schemes, and is a [[full subcategory]] of the category of Durov schemes.

[[Alain Connes]] and [[Caterina Consani]] developed both Soulé and Deitmar's notions by "gluing" the category of multiplicative monoids and the category of rings to create a new category <math>\mathfrak{M}\mathfrak{R},</math> then defining '''F'''₁-schemes to be a particular kind of representable functor on <math>\mathfrak{M}\mathfrak{R}.</math><ref>{{harvtxt|Connes|Consani|2010}}.</ref> Using this, they managed to provide a notion of several number-theoretic constructions over '''F'''₁ such as motives and field extensions, as well as constructing [[Groups of Lie type#Chevalley groups|Chevalley groups]] over '''F'''_1². Along with [[Matilde Marcolli]], Connes-Consani have also connected '''F'''₁ with [[noncommutative geometry]].<ref>{{harvtxt|Connes|Consani|Marcolli|2009}}</ref> It has also been suggested to have connections to the [[unique games conjecture]] in [[computational complexity theory]].<ref>{{citation|url=https://gilkalai.wordpress.com/2018/01/10/subhash-khot-dor-minzer-and-muli-safra-proved-the-2-to-2-games-conjecture/|title=Subhash Khot, Dor Minzer and Muli Safra proved the 2-to-2 Games Conjecture|work=Combinatorics and more|first=Gil|last=Kalai|authorlink=Gil Kalai|date=2018-01-10}}</ref>

== Motivations ==

Lorscheid, along with others, has recently achieved Tit's original aim of describing Chevalley groups over '''F'''₁ by introducing objects called blueprints, which are a simultaneous generalisation of both [[semiring]]s and monoids.<ref>{{harvtxt|Lorscheid|2018a}}</ref><ref>{{harv|Lorscheid|2018b}}</ref> These are used to define so-called "blue schemes", one of which is Spec '''F'''₁.<ref>{{harvtxt|Lorscheid|2016}}</ref> Lorscheid's ideas depart somewhat from other ideas of groups over '''F'''₁, in that the '''F'''₁-scheme is not itself the Weyl group of its base extension to normal schemes. Lorscheid first defines the Tits category, a full subcategory of the category of blue schemes, and defines the "Weyl extension" , a functor from the Tits category to '''Set'''. A Tits-Weyl model of an algebraic group <math>\mathcal{G}</math> is a blue scheme ''G'' with a group operation which is a morphism in the Tits category, whose base extension is <math>\mathcal{G}</math> and whose Weyl extension is isomorphic to the Weyl group of <math>\mathcal{G}.</math>

=== Algebraic number theory ===

One motivation for '''F'''₁ comes from [[algebraic number theory]]. [[André 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''.

'''F'''₁-geometry has been linked to tropical geometry, via the fact that semirings (in particular, tropical semirings) arise as quotients of some monoid semiring '''N'''[''A''] of finite formal sums of elements of a monoid ''A'', which is itself an '''F'''₁-algebra. This connection is made explicit by Lorscheid's use of blueprints.<ref>{{harvtxt|Lorscheid|2015}}</ref> The Giansiracusa brothers have constructed a tropical scheme theory, for which their category of tropical schemes is equivalent to the category of Toën-Vaquié '''F'''₁-schemes.<ref>{{harvtxt|Giansiracusa|Giansiracusa|2016}}</ref> This category embeds faithfully, but not fully, into the category of blue schemes, and is a full subcategory of the category of Durov schemes.

=== Arakelov geometry ===

==Properties==

AnotherGeometry angleover comesa fromfield 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.

'''F'''₁ is expected to have the following properties.

== Expected properties ==

=== F₁ 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.

=== 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>

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*: Given a [[Dynkin diagram]] for a semisimple algebraic group, its [[Weyl group]] is<ref>[http://math.ucr.edu/home/baez/week187.html This Week's Finds in Mathematical Physics, Week 187]</ref> the semisimple algebraic group over '''F'''₁.

* The [[affine scheme]] Spec '''Z''' is a curve over '''F'''₁.

* Groups are [[Hopf algebra]]s over '''F'''₁. More generally, anything defined purely in terms of diagrams of algebraic objects should have an '''F'''₁-{{nbh}}analog in the category of sets.

* [[Group action (mathematics)|Group action]]s on sets are projective representations of ''G'' over '''F'''₁, and in this way, ''G'' is the [[group Hopf algebra]] '''F'''₁[''G''].

* [[Toric variety|Toric varieties]] determine '''F'''₁-{{nbh}}varieties. In some descriptions of '''F'''₁-{{nbh}}geometry the converse is also true, in the sense that the extension of scalars of '''F'''₁-{{nbh}}varieties to '''Z''' are toric.<ref>{{harvtxt|Deitmar|2006}}.</ref> Whilst other approaches to '''F'''₁-{{nbh}}geometry admit wider classes of examples, toric varieties appear to lie at the very heart of the theory.

* The zeta function of '''P'''^''N''('''F'''₁) should be {{nowrap|1=''ζ''(''s'') = ''s''(''s'' − 1)⋯(''s'' − ''N'')}}.<ref name="Soule1999"/>

* The ''m''-th ''K''-{{nbh}}group of '''F'''₁ should be the ''m''-th [[stable homotopy group]] of the [[sphere spectrum]].<ref name="Soule1999"/>

== Computations ==

Various structures on a [[Set (mathematics)|set]] are analogous to structures on a projective space, and can be computed in the same way:

=== Sets are projective spaces ===

The number of elements of {{nowrap|1='''P'''('''F'''{{su|b=''q''|p=''n''|lh=0.9}}) = '''P'''^''n''−1('''F'''_''q'')}}, the {{nowrap|(''n'' − 1)}}-{{nbh}}dimensional [[projective space]] over the [[finite field]] '''F'''_''q'', is the '''[[q-bracket|''q''-{{nbh}}integer]]'''<ref>[http://math.ucr.edu/home/baez/week183.html This Week's Finds in Mathematical Physics, Week 183, ''q''-{{nbh}}arithmetic]</ref>

: <math>[n]_q := \frac{q^n-1}{q-1}=1+q+q^2+\dots+q^{n-1}.</math>

Taking {{nowrap|1=''q'' = 1}} yields {{nowrap|1=[''n'']_''q'' = ''n''}}.

The expansion of the ''q''-{{nbh}}integer into a sum of powers of ''q'' corresponds to the [[Schubert cell]] decomposition of projective space.

=== Permutations are [[Flagmaximal (linear algebra)|flags]] ===

There are ''n''! permutations of a set with ''n'' elements, and [''n'']!_''q''! maximal [[Flag (linear algebra)|flags]] in '''F'''{{su|b=''q''|p=''n''|lh=0.9}}, where

: <math>[n]_q!_q := [1]_q [2]_q \dots [n]_q</math>

is the [[Q-Pochhammer symbol#Relationship to other q-functions|''q''-{{nbh}}factorial]]. Indeed, a permutation of a set can be considered a [[Filtration (mathematics)#Sets|filtered set]], as a flag is a filtered vector space: for instance, the ordering {{nowrap|(0, 1, 2)}} of the set {{nowrap|{{mset|0, 1, 2}}}} corresponds to the filtration {{nowrap|{{mset|0}} ⊂ {{mset|0, 1}} ⊂ {{mset|0, 1, 2}}}}.

=== Subsets are subspaces ===

The [[binomial coefficient]]

: <math>\frac{n!}{m!(n-m)!}</math>

gives the number of ''m''-element subsets of an ''n''-element set, and the [[Q-factorial#Relationship to the q-bracket and the q-binomial|''q''-{{nbh}}binomial coefficient]]

: <math>\frac{[n]_q!_q}{[m]_q!_q[n-m]_q!_q}</math>

gives the number of ''m''-dimensional subspaces of an ''n''-dimensional vector space over '''F'''_''q''.

The expansion of the ''q''-{{nbh}}binomial coefficient into a sum of powers of ''q'' corresponds to the [[Schubert cell]] decomposition of the [[Grassmannian]].

== Monoid schemes ==

Deitmar's construction of monoid schemes<ref>{{harvtxt|Deitmar|2005}}</ref> has been called "the very core of '''F'''₁-{{nbh}}geometry",<ref name=":0">{{harvtxt|Lorscheid|20182018a}}</ref> as most other theories of '''F'''₁-{{nbh}}geometry contain descriptions of monoid schemes. Morally, it mimicks the theory of [[Scheme (mathematics)|schemes]] developed in the 1950s and 1960s by replacing [[commutative ring]]s with [[monoid]]s. The effect of this is to "forget" the additive structure of the ring, leaving only the multiplicative structure. For this reason, it is sometimes called "non-additive geometry".

=== Monoids ===

A '''multiplicative monoid''' is a monoid {{math |''A''}} whichthat also contains an [[absorbing element]] 0 (distinct from the identity 1 of the monoid), such that {{math nowrap|1=0''a'' {{=}} 0}} for every {{math |''a''}} in the monoid {{math |''A''.}} The field with one element is then defined to be {{math nowrap|1='''F'''₁ {{=}} {{mset|0, 1},}}}, the multiplicative monoid of the field with two elements, which is [[initial object|initial]] in the category of multiplicative monoids. A '''monoid ideal''' in a monoid {{math |''A''}} is a subset {{math |''I''}} whichthat is multiplicatively closed, contains 0, and such that {{math nowrap|1=''IA'' {{=}} {{mset|''ra'' : ''r'' ∈ ''I'', ''a'' ∈ ''A''}} {{=}} ''I''.}}. Such an ideal is '''prime''' if <math>{{nowrap|''A\setminus'' ∖ ''I</math>''}} is multiplicatively closed and contains 1.

For monoids {{math |''A''}} and {{math |''B'',}} a '''monoid homomorphism''' is a function {{math nowrap|''f'' : ''A'' → ''B''}} such that;

* {{<math |''>f''(0) {{=}} 0}} ;</math>

* {{<math |''>f''(1) {{=}} 1 ,}}</math> and

* {{<math |''>f''(''ab'') {{=}} ''f''(''a'')''f''(''b'')}}</math> for every {{<math |''>a''}}</math> and {{<math |''>b''}}</math> in {{<math |''>A''.}} .</math>

=== Monoid schemes ===

The ''spectrum'' of a monoid {{math |''A'',}} denoted {{math nowrap|Spec ''A'',}}, is the set of [[prime idealsideal]]s of {{math |''A''.}} The spectrum of a monoid can be given a [[Zariski topology]], by defining [[basis (topology)|basic]] [[open setsset]]s

: <math>U_h = \{\mathfrak{p}\in\text{Spec}A:h\notin\mathfrak{p}\},</math>

for each {{math |''h''}} in {{math |''A''.}} A ''monoidal space'' is a topological space along with a [[sheaf (mathematics)|sheaf]] of multiplicative monoids called the ''structure sheaf''. An ''[[affine monoid]] scheme'' is a monoidal space whichthat is isomorphic to the spectrum of a monoid, and a '''monoid scheme''' is a sheaf of monoids whichthat has an open cover by affine monoid schemes.

Monoid schemes can be turned into ring-theoretic schemes by means of a '''base extension''' [[functor]] <math>-\otimes_{\mathbf{nowrap|– ⊗<sub>'''F}_1}\mathbf{Z}'''_1</mathsub> '''Z'''}} whichthat sends the monoid ''A'' to the '''Z'''-{{nbh}}module (i.e. ring) <math>\mathbf{{nowrap|'''Z}'''[''A''] /\langle 0_A\rangle,{{angle bracket|0<sub>''A''</mathsub>}}}}, and a monoid homomorphism {{math nowrap|''f'' : ''A'' → ''B''}} extends to a ring homomorphism <math>f_{\mathbf{nowrap|''f''_'''Z}}''' : ''A\otimes_{\mathbf{'' ⊗_{'''F}_1}\mathbf{'''₁} '''Z}\to''' → ''B\otimes_{\mathbf{'' ⊗<sub>'''F}_1}\mathbf{Z}'''₁</mathsub> which'''Z'''}} that is linear as a '''Z'''-{{nbh}}module homomorphism. The base extension of an affine monoid scheme is defined via the formula

: <math>\operatorname{Spec}(A)\times_{\operatorname{Spec}(\mathbf{F}_1)}\operatorname{Spec}(\mathbf{Z})=\operatorname{Spec}\big( A\otimes_{\mathbf{F}_1}\mathbf{Z}\big),</math>

which in turn defines the base extension of a general monoid scheme.

=== Consequences ===

This construction achieves many of the desired properties of '''F'''₁-{{nbh}}geometry: {{math nowrap|Spec '''F'''₁}} consists of a single point, so behaves similarly to the spectrum of a field in conventional geometry, and the category of affine monoid schemes is dual to the category of multiplicative monoids, mirroring the duality of affine schemes and commutative rings. Furthermore, this theory satisfies the combinatorial properties expected of '''F'''₁ mentioned in previous sections; for instance, projective space over {{math |'''F'''₁}} of dimension {{math|''n''}} as a monoid scheme is identical to an apartment of projective space over {{math |'''F'''_''q''}} of dimension {{math|''n''}} when described as a building.

However, monoid schemes do not fulfill all of the expected properties of a theory of '''F'''₁-{{nbh}}geometry, as the only varieties whichthat have monoid scheme analogues are [[toric variety|toric varieties]].<ref>{{harvtxt|Deitmar|2006}}</ref> More precisely, if {{math |''X''}} is a monoid scheme whose base extension is a [[Flat morphism|flat]], [[Glossary of algebraic geometry#S|separated]], [[Connected space|connected]] scheme of [[Finite morphism#Morphisms of finite type|finite type]], then the base extension of {{math |''X''}} is a toric variety. Other notions of '''F'''₁-{{nbh}}geometry, such as that of Connes–Consani,<ref>{{harvtxt|Connes|Consani|2010}}</ref> build on this model to describe '''F'''₁-{{nbh}}varieties whichthat are not toric.

== Field extensions ==

One may define [[field extension]]s of the field with one element as the group of [[roots of unity]], or more finely (with a geometric structure) as the [[group scheme of roots of unity]]. This is non-naturally isomorphic to the [[cyclic group]] of order ''n'', the isomorphism depending on choice of a [[primitive root of unity]]:<ref>Mikhail Kapranov, linked at The F_un folklore</ref>

: <math>\mathbf{F}_{1^n} = \mu_n.</math>

Thus a vector space of dimension ''d'' over '''F'''_1^''n'' is a finite set of order ''dn'' on which the roots of unity act freely, together with a base point.

From this point of view the [[finite field]] '''F'''_''q'' is an algebra over '''F'''_1^''n'', of dimension {{nowrap|1=''d'' = (''q'' − 1)/''n''}} for any ''n'' that is a factor of {{nowrap|''q'' − 1}} (for example {{nowrap|1=''n'' = ''q'' − 1}} or {{nowrap|1=''n'' = 1}}). This corresponds to the fact that the group of units of a finite field '''F'''_''q'' (which are the {{nowrap|''q'' − 1}} non-zero elements) is a cyclic group of order {{nowrap|''q'' − 1}}, on which any cyclic group of order dividing {{nowrap|''q'' − 1}} acts freely (by raising to a power), and the zero element of the field is the base point.

Similarly, the [[real number]]s '''R''' are an algebra over '''F'''_1^''2'', of infinite dimension, as the real numbers contain ±1, but no other roots of unity, and the complex numbers '''C''' are an algebra over '''F'''_1^''n'' for all ''n'', again of infinite dimension, as the complex numbers have all roots of unity.

From this point of view, any phenomenon that only depends on a field having roots of unity can be seen as coming from '''F'''₁ – for example, the [[discrete Fourier transform]] (complex-valued) and the related [[number-theoretic transform]] ('''Z'''/''n'''''Z'''-{{nbh}}valued).

== See also ==

* [[Arithmetic derivative]]

* [[Semigroup with one element]]

== Notes ==

{{Reflist|2reflist}}

== Bibliography ==

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* {{citation | editor1-last=Consani | editor1-first=Caterina | editor2-last=Connes | editor2-first=Alain | editor2-link=Alain Connes | title=Noncommutative geometry, arithmetic, and related topics. Proceedings of the 21st meeting of the Japan-U.S. Mathematics Institute (JAMI) held at Johns Hopkins University, Baltimore, MD, USA, March 23–26, 2009 | ___location=Baltimore, MD | publisher=Johns Hopkins University Press | isbn=978-1-4214-0352-6 | year=2011 | zbl=1245.00040 }}

* {{ Citationcitation | last1 = Connes | first1 = Alain | author-link = Alain Connes | last2 = Consani | first2 = Caterina | last3 = Marcolli | first3 = Matilde | author3-link=Matilde Marcolli | title = Fun with <math>\mathbb{F}_1</math> | journal = Journal of Number Theory | volume = 129 | issue = 6 | year = 2009 | arxiv = 0806.2401 | pages=1532–1561 |zbl=1228.11143 | mr=2521492 | doi = 10.1016/j.jnt.2008.08.007 | s2cid = 5327852 }}

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== External links ==

* [[John Baez]]'s This Week's Finds in Mathematical Physics: [http://math.ucr.edu/home/baez/week259.html Week 259]

* [http://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html The Field With One Element] at the ''n''-{{nbh}}category cafe

* [http://sbseminar.wordpress.com/2007/08/14/the-field-with-one-element/ The Field With One Element] at Secret Blogging Seminar

* [http://www.neverendingbooks.org/looking-for-f_un Looking for F_un] and [http://www.neverendingbooks.org/the-f_un-folklore The F_un folklore], Lieven le Bruyn.

* [http://frontarxiv.math.ucdavis.eduorg/abs/0909.0069 Mapping F_1-'''F'''₁{{nbh}}land: An overview of geometries over the field with one element], Javier López Peña, Oliver Lorscheid

* [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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