Field with one element

In mathematics, the field with one element is a suggestive name for an object that "should" exist: many objects in math have properties analogous to objects over a field with $$q elements, where $$q = 1, and the analogy between number fields and function fields is stronger if one includes a field with one element. [ [http://www.ihes.fr/%7Esoule/f1-soule.pdf On the field with one element, by Christophe Soulé] ] [ [http://arxiv.org/abs/math/0608179 F1-schemes and toric varieties, by Anton Deitmar] ]

An actual field with one element does not exist (the axioms of a field assume 0 ≠ 1, and even if they didn't, the zero ring (the ring with a single element) does not have the desired properties), but generalizations of fields do exist which have the required properties, for instance as a particular monad: [ [http://arxiv.org/abs/0704.2030 New Approach to Arakelov Geometry, by Nikolai Durov] ] :The ‘field with one element’ is the free algebraic monad generated by one constant (p.26), or the universal generalized ring with zero (p.33)

The idea of a field with one element goes back at least to Jacques Tits in 1957. [ [http://www.dcorfield.pwp.blueyonder.co.uk/2005/11/november-1-12.html David Corfield, Philosophy of Real Mathematics, 8 November 2005.] ]

This object is denoted as $$mathbf{F}_1.

Philosophy

Under the philosophy of "the field with one element":
* Fields $$mathbf{F}_q are quantum deformations of $$mathbf{F}_1, where $$q is the deformation.
* Finite sets are projective spaces over $$mathbf{F}_1
* Pointed sets [ [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.] ] are vector spaces over $$mathbf{F}_1
* Finite sets are affine spaces over $$mathbf{F}_1
* Coxeter groups are simple algebraic groups over $$mathbf{F}_1
* $$mbox{Spec},mathbf{Z} is [ [http://arxiv.org/abs/math/0608179 F1-schemes and toric varieties, by Anton Deitmar] ] a curve over $$mathbf{F}_1
* Groups are Hopf algebras over $$mathbf{F}_1; indeed, for anything categorically defined over both sets and modules, the set-theoretic concept is the $$mathbf{F}_1-analog
* Group actions ("G"-sets) are projective representations of "G" over $$mathbf{F}_1 (this agrees with the previous: "G" is the group Hopf algebra $$mathbf{F}_1 [G] )

Connections

* Given a Dynkin diagram for a simple algebraic group, its Weyl group is [ [http://math.ucr.edu/home/baez/week187.html This Week's Finds in Mathematical Physics, Week 187] ] the simple algebraic group over $$mathbf{F}_1

Computations

Various structures on a set are analogous to structures on a projective space, and can be computed in the same way:

; Points are projective spaces : The number of elements of $$mathbf{P}(mathbf{F}_q^n)=mathbf{P}_q^{n-1}, the $$(n-1)-dimensional projective space over the "n"-dimension vector space over the finite field $$mathbf{F}_q is the "q"-integer [ [http://math.ucr.edu/home/baez/week183.html This Week's Finds in Mathematical Physics, Week 183, "q"-arithmetic] ] :$$ [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.

; Orders are flags : There are n! orders of a set, and $$ [n] _q! maximal flags in $$mathbf{F}_q^n, where $$ [n] _q! := [1] _q [2] _q dots [n] _q is the q-factorial.

; Subsets are subspaces : There are $$n!/m!(n-m)! "m"-element subsets of an "n" element set, and $$ [n] _q!/ [m] _q! [n-m] _q! "m"-dimensional subspaces of $$mathbf{F}_q^n. The number $$ [n] _q!/ [m] _q! [n-m] _q! is called a q-binomial coefficient.

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

References

External links

* [http://www.ihes.fr/IHES/Scientifique/soule/ Conference at IHES on algebraic geometry over $$mathbf{F}_1]
* [http://arxiv.org/abs/0704.2030 New Approach to Arakelov Geometry, by Nikolai Durov] : constructs a generalized theory of rings and schemes, including $$mathbf{F}_1 and other "exotic" objects.
*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 F_un] and [http://www.neverendingbooks.org/index.php/the-f_un-folklore.html The F_un folklore] , Lieven le Bruyn.