Field with one element

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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, Fun.[1] 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[edit]

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 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.[3] He introduced extensions of F1 and used them to handle 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[4] as Fq 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.[5] 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.[6] Deitmar suggested that F1 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 F1 using symmetric monoidal categories.[8] Nikolai Durov constructed F1 as a commutative algebraic monad.[9] Soulé constructed it using algebras over the complex numbers and functors from categories of certain rings.[10] Borger used descent to construct it from the finite fields and the integers.[11]

Recently, Alain Connes, Caterina Consani and Matilde Marcolli have connected F1 with noncommutative geometry.[12]

Properties[edit]

F1 is expected to have the following properties.

Computations[edit]

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[edit]

The number of elements of P(Fn
q
) = Pn−1(Fq), the (n − 1)-dimensional projective space over the finite field Fq, is the q-integer[16]

[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[edit]

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 permutation (0, 1, 2) corresponds to the filtration {0} ⊂ {0,1} ⊂ {0,1,2}.

Subsets are subspaces[edit]

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 Fq.

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[edit]

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:[17]

\mathbf{F}_{1^n} = \mu_n.

Thus a vector space of dimension d over Fn
1
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 Fq is an algebra over Fn
1
, 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 Fq (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 F2
1
, of infinite dimension, as the real numbers contain ±1, but no other roots of unity, and the complex numbers C are an algebra over Fn
1
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[edit]

Notes[edit]

Bibliography[edit]

  • 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 F1", Journal of Number Theory: 1532–1561 zbl=1228.11143, arXiv:0806.2401 
  • Deitmar, Anton (2005), "Schemes over F1", in van der Geer, G.; Moonen, B.; Schoof, R., Number Fields and Function Fields: Two Parallel Worlds, Progress in Mathematics 239 
  • Deitmar, Anton (2006), F1-schemes and toric varieties, arXiv:math/0608179 
  • Durov, Nikolai (2008), New Approach to Arakelov Geometry, arXiv:0704.2030 
  • Kapranov, Michail; Smirnov, Alexander (1995), Cohomology determinants and reciprocity laws: number field case 
  • Le Bruyn, Lieven (2009), (non)commutative f-un geometry, arXiv:0909.2522 
  • Lescot, Paul (2009), Algebre absolue 
  • 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 
  • Manin, Yuri (1995), "Lectures on zeta functions and motives (according to Deninger and Kurokawa)", Astérisque 228 (4): 121–163 
  • Smirnov, Alexander (1992), "Hurwitz inequalities for number fields", Algebra I Analiz 4 (2): 186–209 
  • Soulé, Christoph (2008), 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 

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