Introduction E9F5FC

Questions FFFFC0

• View
• Edit
• History
• Print

See: Math, Binomial theorem, Finite fields

Field with one element, {$F_1$}

The field with one element, {$F_1$}, is a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case.

My impression is that it relates to my concept of a God who goes beyond himself into himself, who asks, Is God necessary? Would I be even if I wasn't?

The Barratt–Priddy theorem says that the stable homotopy groups of the spheres can be expressed in terms of the plus construction applied to the classifying space of the symmetric group, leading to an identification of K-theory of the field with one element with stable homotopy groups.

• Deitmar, Anton (2006), "Remarks on zeta functions and K-theory over F1", Japan Academy. Proceedings. Series A. Mathematical Sciences, 82 (8): 141–146, https://arxiv.org/abs/math/0605429 , doi:10.3792/pjaa.82.141, ISSN 0386-2194, MR 2279281.

Alain Connes: Characteristic 0 is not that 0=1, but rather, that 1+1=1.

Readings

Introductory

• John Baez on the field with one element
• The field with one element. Peter Cameron.
• The Field With One Element Noah Snyder about Mike Shulman's class
• Geometry and the Absolute Point
• Wikipedia: Field with one element
• F_un folklore Lieven Le Bruyn
• F_un with Manin Lieven Le Bruyn
• See Lieven Le Bruyn: Absolute
• 影片: Alain Connes. The Arithmetic Site. Relates topos, tropical semiring, F1.
• Alain Connes, Caterina Consani. The Arithmetic Site.

Related ideas

• Tropical geometry
• Tropical semiring
• Frobenius endomorphism
• Wheel theory defining an extra point 0/0

Overview

• Mapping F1-land: An overview of geometries over the field with one element Preprint. J.L. Pena, O. Lorscheid.
• F_un Mathematics Lieven Le Bruyn
• Recently, there have been a few approaches to "geometry over F1", such as Borger rBo09s, Connes Consani rCC09s,rCC14s, Durov rDus, Lorscheid rLo12s, Soule rSs, Takagi rTak12s, Töen- Vaquie rTVs and Haran rH07s and rH09s. For relations between these see rPLs.

General theories

• Cyclotomy and Analytic Geometry over {$F_1$}
• Projective geometry over {$F_1$} and the Gaussian binomial coefficients, Henry Cohn, his research
• New Approach to Arakelov Geometry Nikolai Durov.
• New foundations for geometry: Two non-additive languages for arithmetical geometry Shai Haran. This point of view of the general-linear-group suggests also that for the field with one element F we have {$"GL_n(\mathbb{F}_1)"=S_n$}, the symmetric group, which embeds as a common subgroup of all the finite group {$GL_n(\mathbb{F}_p)$}, p prime (or the "field" {$\mathbb{F}\{\pm1\}$}, with {$"GL_n(\mathbb{F}\{\pm1\})"=\{\pm1\}^n \bowtie S_n$}. (The symmetry group for {$B_n$} and {$C_n$}.)
• Quantum {$F_{un}$}: The q=1 Limit of Galois Field Quantum Mechanics, Projective Geometry & the Field With One Element Chang, Lewis, Minic, Takeuchi. Model of passing from quantum to classical mechanics
• (Non) Commutative F-un Geometry Lieven Le Bruyn
• Do the symmetric functions have a function-field analogue? Draft. Darij Grinberg.
  • Darij Grinberg, Research interests: Carlitz-Witt vectors and function-field symmetric functions.
• Characteristic one, entropy and the absolute point. Alain Connes, Caterina Consani.

Special aspects

• Video: Taylor Dupuy: The field with one element and the Riemann Hypothesis
• The Action of Young Subgroups on the Partition Complex, Gregory Z. Arone, D. Lukas B. Brantnerthe. Partition complexes can be thought of as Bruhat-Tits buildings over “the field with one element”. In this heuristic picture, Young subgroups correspond to parabolic subgroups. Complementary collapse then has a neat analogous consequence for parabolic restrictions of Bruhat-Tits buildings.
• Carlitz extensions Keith Conrad.
• Cohomology Determinants and Reciprocity Laws: Number Field Case. M.Kapranov, A.Smirnov.

Interpretations of mathematical structures in terms of {$F_1$}.

• A pointed set may be seen as a vector space over the field with one element. Shai Haran. Non-additive geometry.
• A finite set can be thought of as a finite dimensional vector space over the field with one element. But no such field exists!
• The “general linear group” in n dimensions over the field of one element is the symmetric group {$S_n$}.
• A group is a Hopf algebra over the field with one element.
  • the field K is replaced by the 1-point set
  • there is a natural counit (map to 1 point)
  • there is a natural comultiplication (the diagonal map)
  • the unit is the identity element of the group
  • the multiplication is the multiplication in the group
  • the antipode is the inverse
• The analogy is stronger: Weyl groups, a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other. Orthogonal group The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group,[note 1] and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension)....
• John Baez. Week 187. Starting with a Dynkin diagram and choosing a field we get a simple Lie group (if the field is the real or complex numbers) but more generally, a simple algebraic group, as we do for {$F_q$}. From the Dynkin diagram we can also get the Coxeter group, which behaves as if {$q=1$}.
• Clifford algebra Clifford algebras may be thought of as quantizations (cf. Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.

New Approach to Arakelov Geometry Nikolai Durov.

• A theory of generalized rings and schemes.
• Page 26: {$\mathbb{F}_1=\mathbb{F}_{\varnothing}\langle 0^{[0]}\rangle$} is the free algebraic monad generated by one constant. {$\mathbb{F}_1$}-Mod is the category of sets X with one pointed element {$0_X ∈ X$}. {$\mathbb{F}_1$}-homomorphisms are just maps of pointed sets, i.e. maps {$f: X → Y$}, such that {$f(0_X) = 0_Y$}. We can describe sets {$\mathbb{F}_1(n)$} explicitly: any such set consists of {$n + 1$} elements, namely, n basis elements {$\{k\}_n$}, and a constant {$0$}. [Compare the latter with the basis for a Clifford algebra.] Considers the geometry over the field with one element: ("schemes over {$\textrm{Spec}\;\mathbb{F}_1$}").
• Algebraic monads are non-commutative generalized rings.
• An algebraic monad over Sets is essentially the same thing as an algebraic system.
  • In some sense algebraic systems are something like presentations (by a system of generators and relations) of algebraic monads. Thus different (but equivalent) algebraic systems may correspond to isomorphic algebraic monads, and algebraic monads provide an invariant way of describing algebraic systems. In this way the study of algebraic monads might be thought of as the study of algebraic systems from a categorical point of view, i.e. a categorical approach to universal algebra.
  • Any algebraic endofunctor Σ : Sets → Sets is completely determined by its restriction Σ|N : N → Sets. In fact, this restriction functor Σ 7 → Σ|N induces an equivalence between the category of algebraic endofunc- tors Aalg ⊂ A = Endof (Sets) and Funct(N, Sets) = SetsN. This means thatan algebraic endofunctor Σ is essentially the same thing as a countable collection of sets {Σ(n)}n≥0, together with maps Σ(φ) : Σ(n) → Σ(m), defined for any φ : n → m, such that Σ(ψ ◦ φ) = Σ(ψ) ◦ Σ(φ) and Σ(idn) = idΣ(n). 0.4.3. (Algebraic monads.) On the other hand, if two endofunctors Σ and Σ′ commute with filtered inductive limits, the same is true for their composition Σ⊗Σ′ = Σ◦Σ′, i.e. Aalg ∼= SetsN is a full ⊗-subcategory of A = Endof (Sets). Therefore, we can define an algebraic monad Σ = (Σ, μ, ε) as an algebra in A_{alg}. Of course, an algebraic monad is just a monad, such that its underlying endofunctor commutes with filtered inductive limits.
• nLab: Finitary monad is the same as algebraic monad. An algebraic monad is a monoid in the category of algebraic endofunctors on Set.

F1-believers base their f-unny intuition on the following two mantras :

• F1 forgets about additive data and retains only multiplicative data.
• F1-objects only acquire flesh when extended to Z (or C).

What form does the binomial theorem take in a noncommutative ring? In general one can say nothing interesting, but certain special cases work out elegantly. One of the nicest, due to Schutzenberger [18], deals with variables x, y, and q such that q commutes with x and y, and yx = qxy.

Of course, there is no field F1 with only one element, but there is a trivial ring, and it is merely a convention that we do not call it a field. However, it is an excellent convention, because the trivial ring has no nontrivial modules (if x is an element of a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1, since F n 1 does not depend on n. I know of no direct solution to this puzzle, nor of any way to make sense of vector spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes much easier to understand when it is reformulated in terms of projective geometry. That may not be surprising, if one keeps in mind that many topics, such as intersection theory, become simpler when one moves to projective geometry. (The papers [11] and [22] also shed light on this puzzle by indirect routes, but not by using projective geometry.) Cohn, page 489.

Projective line on {$F_1$}

• The projective line over a finite field Fq of q elements has q + 1 points. In all other respects it is no different from projective lines defined over other types of fields. In the terms of homogeneous coordinates [x : y], q of these points have the form: [a : 1] for each a in Fq, and the remaining point at infinity may be represented as [1 : 0].
• Thus the projective line over {$F_1$} of 1 element has 2 points. In terms of homogeneous coordinates, one of these points has the form [a : 1] where {$a\in F_1$}, and the other point at infinity may be represented as [1:0].
• The projective line over {$F_2$} has 3 points: [0:1], [1:1], [1:0] where the latter is the point at infinity.
• The projective line over {$F_3$} has 4 points: [0:1], [1:1], [2:1], [1:0] where the latter is the point at infinity.
• Projective line over {$F_1$} has two points. The second points is infinity. So what does it mean to say {$0=1=\infty$}?

Ideas

Cognitive ideas regarding {$F_1$}.

• Limit as q->1
  • The field with one element can be imagined by way of Pascal's triangle. For the Gaussian binomial coefficients count the number of subspaces of a vector space over a field of characteristic q. Having q->1 yields the usual binomial coefficients. Also, Pascal's triangle counts the simplexes of a subsimplex, and variants count the parts of other infinite families of polytopes.
• I believe that the field with one element can be interpreted as an element that can be interpreted as 0, 1 and infinity. For example, 0 * infinity = 1. I think that 1 can be thought of as reflecting 0 and infinity in a duality. I think of this as modeling God's dance.
• Relation of list to set. List/Set.
• A field relates two groups: an additive group (the level) and a multiplicative group (the metalevel of actions). As regards the action, the zero of the additive group is the negation of action - no action taken, whereas the one of the multiplicative group is the action that has no effect. Therein lies the distinction of the level and the metalevel.
• I think that an affine geometry is not so much distinguished by its not having a zero (a zero or origin can always be defined) but by its not having a one. Perhaps a projective geometry has both a zero and an infinity and so a one is naturally available.
• Zero is not a choice. The field needs to offer another choice.
• There exists (there is one=1) a unique (and only one) vs. For any (all=infinity) vs. There is none (negation=0).
• Fields are "complete" mathematical structures (having all of the operations) but thus inevitably having a "gap" by which 0 and 1 are distinct. This is the quintessential gap and the prime numbers are likewise gaps in the factorization of numbers.
• P and NP. Field with 1 element - deterministic. Char q - nondeterministic.
• Intersection and union do not have inverses.
• Intersection and union are dual. they are distributive over each other. addition and multiplication are similar but addition is not distributive over multtiplication.
• The anharmonic group (see Cross-ratio) permutes 0,1 and infinity.
• One choice is the same as no choice. {$F_1$}
• The theory of fields is unlike the theories of groups, rings, etc. because the inverse operation is not defined for all x (not defined for x=0). Thus there is no forgetful functor from Field to Set. This is relevant for {$F_1$}.

Finite fields

• Lyndon words - irreducible polynomials for finite fields
• Duality of q and n in {$GL_n(F_q)$}.
• Multiset of Lyndon words - reducible and irreducible. Homogeneous symmetric functions of eigenvalues.
• Interpolation between homogeneous and elementary - between commutativity and anti-commutativity.
• Lyndon words are like prime numbers.
• Dimension of free Lie algebras = number of Lyndon words of length n
• What would be the q-theory for finite fields for matrix combinatorics?

Dear Harvey,

Thank you for your invitations in your letter below and also earlier, "...I am trying to get a dialog going on the FOM and in these other forums as to "what foundations of mathematics are, ought to be, and what purpose they serve"." http://www.cs.nyu.edu/pipermail/fom/2016-April/019724.html

You mentioned, in my words, that you are looking for an issue that working mathematicians are grappling with where the classical ZFC foundations are not satisfactory or sufficient. Would the "field with one element" be such issue for you? https://ncatlab.org/nlab/show/field+with+one+element

Jacques Tits raised this issue in 1957 and it has yet to be resolved despite substantial interest, conferences, and long papers. Would that count as a "problem" for the Foundations of Mathematics? It seems that in the history of math it is very easy to simply say "that is not real math" as was the case with the rational numbers, imaginary numbers, infinitesimals, infinite series, etc.

The issue is that there are many instances where a combinatorial interpretation makes sense in terms of a finite field Fq of characteristic q, which is all the more insightful when q=1. For example, the Gaussian binomial coefficients can be interpreted as counting the number of k-dimensional subspaces of an n-dimensional vector space over a finite field Fq. When q=1, then we get the usual binomial coefficients which count the subsets of size k of a set of size n. So this suggests an important way of thinking about sets. However, F1 would be a field with one element, which means that 0=1. But if 0 and 1 are not distinct, then none of the usual properties of a field make sense. Nobody has figured out a convincing interpretation for F1. And yet the concept seems to be pervasive, meaningful and fruitful.

If there was an alternate foundations of mathematics which yielded a helpful, meaningful, fruitful interpretation of F1, would that count in its favor? And if it could do everything that FOM can do, then might it be preferable, at least for some? But especially if that interpretation was shown not to make sense in other FOMs?

David Corfield's post https://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html about Nikolai Durov's book http://arxiv.org/abs/0704.2030 John Baez: This fits nicely with my own intuitions about linear algebra over the field with one element. A pointed set acts like a ‘vector space over the field with one element’; a set acts like a projective space over the field with one element.

• http://math.ucr.edu/home/baez/week185.html
• http://math.ucr.edu/home/baez/week186.html
• http://math.ucr.edu/home/baez/week187.html

Thomas Krantz: Could there be only one truth value, i.e. instead of {true,false} only one: true-false so to say? It is related to geometry based on the one element field F_1 which is very studied these days. Related to Jean-Yves Girard. Linear Logic: Its Syntax and Semantics

Consider combinatorial interpretations of the Gaussian binomial coefficients

• Weights on simplexes
• Inversions, r-combinations
• r indistinguishable balls into m indistinguishable bins
• analogs of Pascal's triangle, Young tableaux

Thomas noted the symmetry of {$x^0=1$}. Relate this to {$F_1$}, choosing one out of one, or none out of none.

• Choosing one out of one: Driving on a winding road, each turn is a choice of one out of one. Whereas a fork is a choice of one out of two, a usual intersection is a choice of one out of three and so on.

John Baez. Mathematics in the 21st-Century. Slides. Field with one element. Commutative algebra {$^{op}\cong$} Geometry