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.