Field with one element

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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[1][2].

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[3]:

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[4].

We denote this object \mathbf{F}_1.

Contents

[edit] 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[5] 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[6] 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])

[edit] Connections

[edit] 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[8]
[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.

[edit] References

  1. ^ On the field with one element, by Christophe Soulé
  2. ^ F1-schemes and toric varieties, by Anton Deitmar
  3. ^ New Approach to Arakelov Geometry, by Nikolai Durov
  4. ^ David Corfield, Philosophy of Real Mathematics, 8 November 2005.
  5. ^ [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.]
  6. ^ F1-schemes and toric varieties, by Anton Deitmar
  7. ^ This Week's Finds in Mathematical Physics, Week 187
  8. ^ This Week's Finds in Mathematical Physics, Week 183, q-arithmetic

[edit] External links