en>ChrisGualtieri: General Fixes using AWB

2013-10-20T14:13:41Z

General Fixes using AWB

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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. 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, most 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 {{Harv|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''-dimensional abstract simplicial complex, and if {{nowrap|''k'' < ''n''}}, then every ''k''-simplex of the building must be contained in at least three ''n''-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 which 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.

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.

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.

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 '''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''.

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 [[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> [[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>

Recently, [[Alain Connes]], Caterina Consani and Matilde Marcolli have connected '''F'''1 with [[noncommutative geometry]].<ref>{{harvtxt|Connes|Consani|Marcolli|2009}}.</ref>

==Properties==
'''F'''1 is expected to have the following 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>
* The finite fields '''F'''''q'' are [[quantum group|quantum deformations]] of '''F'''1, where ''q'' is the deformation.
* [[Weyl group]]s are simple algebraic groups over '''F'''1:
*: 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-analog in the category of sets.
* [[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-varieties. In some descriptions of '''F'''1-geometry the converse is also true, in the sense that the extension of scalars of '''F'''1-varieties to '''Z''' are toric<ref>{{harvtxt|Deitmar|2006}}.</ref> Whilst other approaches to '''F'''1-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''-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 '''P'''('''F'''{{su|b=''q''|p=''n''}}) = '''P'''''n''−1('''F'''''q''), the {{nowrap|(''n'' − 1)}}-dimensional [[projective space]] over the [[finite field]] '''F'''''q'', is the '''[[q-bracket|''q''-integer]]'''<ref>[http://math.ucr.edu/home/baez/week183.html This Week's Finds in Mathematical Physics, Week 183, ''q''-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''-integer into a sum of powers of ''q'' corresponds to the [[Schubert cell]] decomposition of projective space.

===Permutations are [[Flag (linear algebra)|flags]]===
There are ''n''! permutations of a set with ''n'' elements, and [''n'']''q''! maximal flags in '''F'''{{su|b=''q''|p=''n''}}, where
:<math>[n]_q! := [1]_q [2]_q \dots [n]_q</math>
is the [[Q-factorial#Relationship to the q-bracket and the q-binomial|''q''-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 permutation {{nowrap|(0, 1, 2)}} corresponds to the filtration {0} ⊂ {0,1} ⊂ {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''-binomial coefficient]]
:<math>\frac{[n]_q!}{[m]_q![n-m]_q!}</math>
gives the number of ''m''-dimensional subspaces of an ''n''-dimensional vector space over '''F'''''q''.

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

==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'''{{su|b=1|p=''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'''{{su|b=1|p=''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'''{{su|b=1|p=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'''{{su|b=1|p=''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'''-valued).

==See also==
* [[Arithmetic derivative]]
* [[Semigroup with one element]]

==Notes==
{{Reflist|2}}

==Bibliography==
* {{ Citation | last1 = Borger | first1 = James | title = Λ-rings and the field with one element | year = 2009 | arxiv=0906.3146 }}
* {{ Citation | last1 = Connes | first1 = Alain | author-link = Alain Connes | last2 = Consani | first2 = Caterina | last3 = Marcolli | first3 = Matilde | title = Fun with '''F'''1 | journal = Journal of Number Theory | year = 2009 | arxiv = 0806.2401 }}
* {{ Citation | last1 = Deitmar | first1 = Anton | chapter = Schemes over '''F'''1 | book = Number Fields and Function Fields: Two Parallel Worlds | editor-last = van der Geer | editor-first = G. | editor2-last = Moonen | editor2-first = B. | editor3-last = Schoof | editor3-first = R. | series = Progress in Mathematics | volume = 239 | year = 2005 }}
* {{ Citation | last1 = Deitmar | first1 = Anton | title = '''F'''1-schemes and toric varieties | year = 2006 | arxiv = math/0608179 }}
* {{ Citation | last1 = Durov | first1 = Nikolai | title = New Approach to Arakelov Geometry | year = 2008 | arxiv = 0704.2030 }}
* {{ Citation | last1 = Kapranov | first1 = Michail | last2 = Smirnov | first2 = Alexander | title = Cohomology determinants and reciprocity laws: number field case | year = 1995 | url = http://www.neverendingbooks.org/DATA/KapranovSmirnov.pdf }}
* {{ Citation | last = Le Bruyn | first = Lieven | title = (non)commutative f-un geometry | year = 2009 | arxiv = 0909.2522 }}
* {{ Citation | last1 = Lescot | first1 = Paul | title = Algebre absolue | year = 2009 | url = http://www.univ-rouen.fr/LMRS/Persopage/Lescot/algabsodef.pdf }}
* {{ Citation | last1 = López Peña | first1 = Javier | last2 = Lorscheid | first2 = Oliver | title = Mapping '''F'''1-land: An overview of geometries over the field with one element | journal = Noncommutative Geometry, Arithmetic, and related topics | pages = 241–265 | year = 2011 | arxiv = 0909.0069 }}
* {{ Citation | title = Algebraic groups over the field with one element | first = Oliver | last = Lorscheid | year = 2009 | arxiv = 0907.3824 }}
* {{ Citation | last1 = Manin | first1 = Yuri | author-link = Yuri Manin | title = Lectures on zeta functions and motives (according to Deninger and Kurokawa) | journal = Astérisque | volume = 228 | issue = 4 | year = 1995 | pages = 121–163 }}
* {{ Citation | last1 = Smirnov | first1 = Alexander | title = Hurwitz inequalities for number fields | journal = Algebra I Analiz | volume = 4 | issue = 2 | year = 1992 | pages = 186–209 }}

* {{ Citation | last1 = Soulé | first1 = Christoph | title = Les variétés sur le corps à un élément | year = 2008 | arxiv = math/0304444 | language = French }}
* {{ Citation | last1 = Tits | first1 = Jacques | chapter = Sur les analogues algébriques des groupes semi-simples complexes | title = Colloque d’algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques Établissements Ceuterick, Louvain | publisher = Librairie Gauthier-Villars | place = Paris | year = 1957 | pages = 261–289 }}
* {{ Citation | last1 = Toën | first1 = Bertrand | last2 = Vaquié | first2 = Michel | title = Au dessous de Spec '''Z''' | arxiv = math/0509684 | year = 2005 }}

==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''-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/index.php/looking-for-f_un.html Looking for Fun] and [http://www.neverendingbooks.org/index.php/the-f_un-folklore.html The Fun folklore], Lieven le Bruyn.
* [http://front.math.ucdavis.edu/0909.0069 Mapping F_1-land:An overview of geometries over the field with one element], Javier López Peña, Oliver Lorscheid
* [http://cage.ugent.be/~kthas/Fun Fun Mathematics], Lieven le Bruyn, [[Thas, Koen|Koen Thas]].
* [http://www.ihes.fr/IHES/Scientifique/soule/ Conference at IHES on algebraic geometry over '''F'''1]
* Vanderbilt conference on [http://www.math.vanderbilt.edu/~ncgoa/workshop2008.html Noncommutative Geometry and Geometry over the Field with One Element] ([http://www.math.vanderbilt.edu/~ncgoa/schedule_workshop08.pdf Schedule])
* [http://noncommutativegeometry.blogspot.com/2008/05/ncg-and-fun.html NCG and F_un], by [[Alain Connes]] and K. Consani: summary of talks and slides

{{DEFAULTSORT:Field With One Element}}
[[Category:Algebraic geometry]]
[[Category:Noncommutative geometry]]
[[Category:Finite fields]]

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en>ChrisGualtieri: General Fixes using AWB