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 F₁, or, in a French–English pun, F_un. The name "field with one element" and the notation F₁ are only suggestive, as there is no field with one element in classical abstract algebra. Instead, F₁ 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.

F₁ 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 F₁ 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 F₁ 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 F₁ 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. F₁ has been connected to noncommutative geometry and to a possible proof of the Riemann hypothesis. Many theories of F₁ have been proposed, but it is not clear which, if any, of them give F₁ 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. Using this analogy it was possible to describe some of the elementary properties of F₁, but it was not possible to construct it.

A separate inspiration for F₁ 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 F₁ is that Z should be an F₁-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 F₁ is useful for technical reasons.

By 1991, Alexander Smirnov had taken some steps towards algebraic geometry over F₁. He introduced extensions of F₁ and used them to handle the projective line P¹ over F₁. Algebraic numbers were treated as maps to this P¹, 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 F₁ later on were denoted as F_q with q = 1ⁿ.

In 1993, Yuri Manin gave a series of lectures on zeta functions where he proposed developing a theory of algebraic geometry over F₁. He suggested that zeta functions of varieties over F₁ would have very simple descriptions, and he proposed a relation between the K-theory of F₁ and the homotopy groups of spheres. This inspired several people to attempt to construct F₁. In 2000, Zhu proposed that F₁ was the same as F₂ except that the sum of one and one was one, not zero. Deitmar suggested that F₁ should be found by forgetting the additive structure of a ring and focusing on the multiplication. Toën and Vaquié built on Hakim's theory of relative schemes and defined F₁ using symmetric monoidal categories. Nikolai Durov constructed F₁ as a commutative algebraic monad. Soulé constructed it using algebras over the complex numbers and functors from categories of certain rings. Borger used descent to construct it from the finite fields and the integers.

Recently, Alain Connes, Caterina Consani and Matilde Marcolli have connected F₁ with noncommutative geometry. It has also been suggested to have connections to the unique games conjecture in computational complexity theory.

Properties

F₁ is expected to have the following properties.

• Finite sets are both affine spaces and projective spaces over F₁.
• Pointed sets are vector spaces over F₁.
• The finite fields F_q are quantum deformations of F₁, where q is the deformation.
• Weyl groups are simple algebraic groups over F₁:

  Given a Dynkin diagram for a semisimple algebraic group, its Weyl group is the semisimple algebraic group over F₁.

• The affine scheme Spec Z is a curve over F₁.
• Groups are Hopf algebras over F₁. More generally, anything defined purely in terms of diagrams of algebraic objects should have an F₁-analog in the category of sets.
• Group actions on sets are projective representations of G over F₁, and in this way, G is the group Hopf algebra F₁[G].
• Toric varieties determine F₁-varieties. In some descriptions of F₁-geometry the converse is also true, in the sense that the extension of scalars of F₁-varieties to Z are toric Whilst other approaches to F₁-geometry admit wider classes of examples, toric varieties appear to lie at the very heart of the theory.
• The zeta function of P^N(F₁) should be ζ(s) = s(s − 1)⋯(s − N).
• The m-th K-group of F₁ should be the m-th stable homotopy group of the sphere spectrum.

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(Fⁿ
_q) = Pⁿ⁻¹(F_q), the (n − 1)-dimensional projective space over the finite field F_q, is the q-integer

$[n]_{q}:={\frac {q^{n}-1}{q-1}}=1+q+q^{2}+\dots +q^{n-1}.$

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 Fⁿ
_q, where

$[n]_{q}!:=[1]_{q}[2]_{q}\dots [n]_{q}$

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

The binomial coefficient

${\frac {n!}{m!(n-m)!}}$

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

${\frac {[n]_{q}!}{[m]_{q}![n-m]_{q}!}}$

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 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:

$\mathbf {F} _{1^{n}}=\mu _{n}.$

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 F₁ – for example, the discrete Fourier transform (complex-valued) and the related number-theoretic transform (Z/nZ-valued).

See also

• Arithmetic derivative
• Semigroup with one element

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 1-4214-0352-8, Zbl 1245.00040
• Connes, Alain; Consani, Caterina; Marcolli, Matilde (2009), "Fun with $\mathbb {F} _{1}$", Journal of Number Theory, 129 (6): 1532–1561, arXiv:0806.2401, doi:10.1016/j.jnt.2008.08.007, MR 2521492, Zbl 1228.11143
• Deitmar, Anton (2005), "Schemes over F₁", in van der Geer, G.; Moonen, B.; Schoof, R. (eds.), Number Fields and Function Fields: Two Parallel Worlds, Progress in Mathematics, vol. 239
• Deitmar, Anton (2006), F₁-schemes and toric varieties, arXiv:math/0608179
• Durov, Nikolai (2008), "New Approach to Arakelov Geometry", arXiv:0704.2030 [math.AG]
• 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 {{citation}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)
• López Peña, Javier; Lorscheid, Oliver (2011), "Mapping F₁-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]
• 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 Z, arXiv:math/0509684

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.
• Conference at IHES on algebraic geometry over F₁
• 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