# Field with one element

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}.[1] 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 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 spaces and polynomial rings 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 Tits 1957, on the basis of an analogy between symmetries in projective geometry and the combinatorics of simplicial complexes. **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 buildings, which relate algebraic groups to abstract simplicial complexes. One of the assumptions is a non-triviality condition: If the building is an *n*-dimensional abstract simplicial complex, and if *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 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*.[2] 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 *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 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 **Z** ×_{F1} **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}.[3] He introduced extensions of **F**_{1} and used them to handle the projective line **P**^{1} over **F**_{1}. Algebraic numbers were treated as maps to this **P**^{1}, and conjectural approximations to the Riemann–Hurwitz formula for these maps were suggested. These approximations imply very profound assertions like the abc conjecture. The extensions of **F**_{1} later on were denoted[4] as **F**_{q} with *q* = 1^{n}.

In 1993, Yuri Manin gave a series of lectures on zeta functions where he proposed developing a theory of algebraic geometry over **F**_{1}.[5] 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.[6] Deitmar suggested that **F**_{1} should be found by forgetting the additive structure of a ring and focusing on the multiplication.[7] Toën and Vaquié built on Hakim's theory of relative schemes and defined **F**_{1} using symmetric monoidal categories[8] Their construction was later shown to be equivalent to Deitmar's by Vezzani.[9] Nikolai Durov constructed **F**_{1} as a commutative algebraic monad.[10] Soulé constructed it using algebras over the complex numbers and functors from categories of certain rings.[11] Borger used descent to construct it from the finite fields and the integers.[12]

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 then defining **F**_{1}-schemes to be a particular kind of representable functor on [13] 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 Chevalley groups over **F**_{12}. Along with Matilde Marcolli, Connes-Consani have also connected **F**_{1} with noncommutative geometry.[14] It has also been suggested to have connections to the unique games conjecture in computational complexity theory.[15]

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 semirings and monoids.[16][17] These are used to define so-called "blue schemes", one of which is Spec **F**_{1}.[18] 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 is a blue scheme *G* with a group operation which is a morphism in the Tits category, whose base extension is and whose Weyl extension is isomorphic to the Weyl group of

**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.[19] 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.[20] This category embeds faithfully, but not fully, into the category of blue schemes, and is a full subcategory of the category of Durov schemes.

## Properties

**F**_{1} is expected to have the following properties.

- Finite sets are both affine spaces and projective spaces over
**F**_{1}. - Pointed sets are vector spaces over
**F**_{1}.[21] - The finite fields
**F**_{q}are quantum deformations of**F**_{1}, where*q*is the deformation. - Weyl groups are simple algebraic groups over
**F**_{1}:- Given a Dynkin diagram for a semisimple algebraic group, its Weyl group is[22] the semisimple algebraic group over
**F**_{1}.

- Given a Dynkin diagram for a semisimple algebraic group, its Weyl group is[22] the semisimple algebraic group over
- The affine scheme Spec
**Z**is a curve over**F**_{1}. - Groups are Hopf algebras 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 actions on sets are projective representations of
*G*over**F**_{1}, and in this way,*G*is the group Hopf algebra**F**_{1}[*G*]. - 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.[23] 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 ζ(*s*) =*s*(*s*− 1)⋯(*s*−*N*).[11] - The
*m*-th*K*-group of**F**_{1}should be the*m*-th stable homotopy group of the sphere spectrum.[11]

## Computations

Various structures on a 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**^{n}_{q}) = **P**^{n−1}(**F**_{q}), the (*n* − 1)-dimensional projective space over the finite field **F**_{q}, is the ** q-integer**[24]

Taking *q* = 1 yields [*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 flags

There are *n*! permutations of a set with *n* elements, and [*n*]_{q}! maximal flags in **F**^{n}_{q}, where

is the *q*-factorial. Indeed, a permutation of a set can be considered a filtered set, as a flag is a filtered vector space: for instance, the ordering (0, 1, 2) of the set {0,1,2} corresponds to the filtration {0} ⊂ {0,1} ⊂ {0,1,2}.

### Subsets are subspaces

gives the number of *m*-element subsets of an *n*-element set, and the *q*-binomial coefficient

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.

## Monoid schemes

Deitmar's construction of monoid schemes[25] has been called "the very core of **F**_{1}-geometry",[26] as most other theories of **F**_{1}-geometry contain descriptions of monoid schemes. Morally, it mimicks the theory of schemes developed in the 1950s and 1960s by replacing commutative rings with monoids. 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 *A* which also contains an absorbing element 0 (distinct from the identity 1 of the monoid), such that 0*a* = 0 for every *a* in the monoid *A*. The field with one element is then defined to be **F**_{1} = {0,1}, the multiplicative monoid of the field with two elements, which is initial in the category of multiplicative monoids. A **monoid ideal** in a monoid *A* is a subset *I* which is multiplicatively closed, contains 0, and such that *IA* = {*ra* : *r*∈*I*, *a*∈*A*} = *I*. Such an ideal is **prime** if is multiplicatively closed and contains 1.

For monoids *A* and *B*, a **monoid homomorphism** is a function *f* : *A* → *B* such that;

*f*(0) = 0;*f*(1) = 1, and*f*(*ab*) =*f*(*a*)*f*(*b*) for every*a*and*b*in*A*.

### Monoid schemes

The *spectrum* of a monoid *A*, denoted Spec *A*, is the set of prime ideals of *A*. The spectrum of a monoid can be given a Zariski topology, by defining basic open sets

for each *h* in *A*. A *monoidal space* is a topological space along with a sheaf of multiplicative monoids called the *structure sheaf*. An *affine monoid scheme* is a monoidal space which is isomorphic to the spectrum of a monoid, and a **monoid scheme** is a sheaf of monoids which has an open cover by affine monoid schemes.

Monoid schemes can be turned into ring-theoretic schemes by means of a **base extension** functor which sends the monoid *A* to the **Z**-module (i.e. ring) and a monoid homomorphism *f* : *A* → *B* extends to a ring homomorphism which is linear as a **Z**-module homomorphism. The base extension of an affine monoid scheme is defined via the formula

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

### Consequences

This construction achieves many of the desired properties of **F**_{1}-geometry: 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 **F**_{1} of dimension *n* as a monoid scheme is identical to an apartment of projective space over **F**_{q} of dimension *n* when described as a building.

However, monoid schemes do not fulfill all of the expected properties of a theory of **F**_{1}-geometry, as the only varieties which have monoid scheme analogues are toric varieties.[27] More precisely, if *X* is a monoid scheme whose base extension is a flat, separated, connected scheme of finite type, then the base extension of *X* is a toric variety. Other notions of **F**_{1}-geometry, such as that of Connes–Consani,[28] build on this model to describe **F**_{1}-varieties which are not toric.

## Field extensions

One may define field extensions 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:[29]

Thus a vector space of dimension *d* over **F**_{1n} 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**_{1n}, of dimension *d* = (*q* − 1)/*n* for any *n* that is a factor of *q* − 1 (for example *n* = *q* − 1 or *n* = 1). This corresponds to the fact that the group of units of a finite field **F**_{q} (which are the *q* − 1 non-zero elements) is a cyclic group of order *q* − 1, on which any cyclic group of order dividing *q* − 1 acts freely (by raising to a power), and the zero element of the field is the base point.

Similarly, the real numbers **R** are an algebra over **F**_{12}, 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**_{1n} 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).

## Notes

- "un" is French for "one", and fun is a playful English word. For examples of this notation, see, e.g. Le Bruyn (2009), or the links by Le Bruyn, Connes, and Consani.
- Tits (1957).
- Smirnov (1992)
- Kapranov & Smirnov (1995)
- Manin (1995).
- Lescot (2009).
- Deitmar (2005).
- Toën & Vaquié (2005).
- Vezzani (2010)
- Durov (2008).
- Soulé (1999)
- Borger (2009).
- Connes & Consani (2010).
- Connes, Consani & Marcolli (2009)
- Kalai, Gil (10 January 2018), "Subhash Khot, Dor Minzer and Muli Safra proved the 2-to-2 Games Conjecture",
*Combinatorics and more* - Lorscheid (2018a)
- (Lorscheid 2018b)
- Lorscheid (2016)
- Lorscheid (2015)
- Giansiracusa & Giansiracusa (2016)
- Noah Snyder, The field with one element, Secret Blogging Seminar, 14 August 2007.
- This Week's Finds in Mathematical Physics, Week 187
- Deitmar (2006).
- This Week's Finds in Mathematical Physics, Week 183,
*q*-arithmetic - Deitmar (2005)
- Lorscheid (2018)
- Deitmar (2006)
- Connes & Consani (2010)
- Mikhail Kapranov, linked at The F_un folklore

## Bibliography

- Borger, James (2009),
*Λ-rings and the field with one element*, arXiv:0906.3146 - Consani, Caterina; Connes, Alain, eds. (2011),
*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*, Baltimore, MD: Johns Hopkins University Press, ISBN 978-1-4214-0352-6, Zbl 1245.00040 - Connes, Alain; Consani, Caterina; Marcolli, Matilde (2009), "Fun with ",
*Journal of Number Theory*,**129**(6): 1532–1561, arXiv:0806.2401, doi:10.1016/j.jnt.2008.08.007, MR 2521492, Zbl 1228.11143 - Connes, Alain; Consani, Caterina (2010), "Schemes over
**F**_{1}and zeta functions",*Compositio Mathematica*, London Mathematical Society,**146**(6): 1383–1415, arXiv:0903.2024, doi:10.1112/S0010437X09004692 - Deitmar, Anton (2005), "Schemes over
**F**_{1}", in van der Geer, G.; Moonen, B.; Schoof, R. (eds.),*Number Fields and Function Fields: Two Parallel Worlds*, Progress in Mathematics,**239** - Deitmar, Anton (2006),
, arXiv:math/0608179**F**_{1}-schemes and toric varieties - Durov, Nikolai (2008), "New Approach to Arakelov Geometry", arXiv:0704.2030 [math.AG]
- Giansiracusa, Jeffrey; Giansiracusa, Noah (2016), "Equations of tropical varieties",
*Duke Mathematical Journal*,**165**(18): 3379–3433, arXiv:1308.0042, doi:10.1215/00127094-3645544 - Kapranov, Mikhail; Smirnov, Alexander (1995),
*Cohomology determinants and reciprocity laws: number field case*(PDF) - Le Bruyn, Lieven (2009), "(non)commutative f-un geometry", arXiv:0909.2522 [math.RA]
- Lescot, Paul (2009),
*Algebre absolue*(PDF), archived from the original (PDF) on 27 July 2011, retrieved 21 November 2009 - López Peña, Javier; Lorscheid, Oliver (2011), "Mapping
**F**_{1}-land: An overview of geometries over the field with one element",*Noncommutative Geometry, Arithmetic, and Related Topics*: 241–265, arXiv:0909.0069 - Lorscheid, Oliver (2009), "Algebraic groups over the field with one element", arXiv:0907.3824 [math.AG]
- Lorscheid, Oliver (2016), "A blueprinted view on
**F**_{1}-geometry", in Koen, Thas (ed.),*Absolute arithmetic and*, European Mathematical Society Publishing House, arXiv:1301.0083**F**_{1}-geometry - Lorscheid, Oliver (2018a), "
**F**_{1}for everyone",*Jahresbericht der Deutschen Mathematiker-Vereinigung*, Springer,**120**(2): 83–116, arXiv:1801.05337, doi:10.1365/s13291-018-0177-x - Lorscheid, Oliver (2018b), "The geometry of blueprints part II: Tits-Weyl models of algebraic groups",
*Forum of Mathematics, Sigma*,**6**, arXiv:1201.1324 - Lorscheid, Oliver (2015),
*Scheme-theoretic tropicalization*, arXiv:1508.07949 - Manin, Yuri (1995), "Lectures on zeta functions and motives (according to Deninger and Kurokawa)",
*Astérisque*,**228**(4): 121–163 - Scholze, Peter (2017),
*p-adic geometry*, p. 13, arXiv:1712.03708 - Smirnov, Alexander (1992), "Hurwitz inequalities for number fields",
*Algebra I Analiz*,**4**(2): 186–209 - Soulé, Christophe (1999),
*On the field with one element (exposé à l'Arbeitstagung, Bonn, June 1999)*(PDF), Preprint IHES - Soulé, Christophe (2003),
*Les variétés sur le corps à un élément*(in French), arXiv:math/0304444 - Tits, Jacques (1957), "Sur les analogues algébriques des groupes semi-simples complexes",
*Colloque d'algèbre supérieure, tenu à Bruxelles du 19 au 22 décembre 1956, Centre Belge de Recherches Mathématiques Établissements Ceuterick, Louvain*, Paris: Librairie Gauthier-Villars, pp. 261–289 - Toën, Bertrand; Vaquié, Michel (2005),
*Au dessous de Spec*, arXiv:math/0509684**Z** - Vezzani, Alberto (2010), "Deitmar's versus Toën-Vaquié's schemes over
**F**_{1}",*Mathematische Zeitschrift*,**271**: 1–16, arXiv:1005.0287, doi:10.1007/s00209-011-0896-5

## External links

- John Baez's This Week's Finds in Mathematical Physics: Week 259
- The Field With One Element at the
*n*-category cafe - The Field With One Element at Secret Blogging Seminar
- Looking for F
_{un}and The F_{un}folklore, Lieven le Bruyn. - Mapping F_1-land:An overview of geometries over the field with one element, Javier López Peña, Oliver Lorscheid
- F
_{un}Mathematics, Lieven le Bruyn, Koen Thas. - Vanderbilt conference on Noncommutative Geometry and Geometry over the Field with One Element (Schedule)
- NCG and F_un, by Alain Connes and K. Consani: summary of talks and slides