# On the metaphysics of F1

Alain Connes and Caterina Consani111Partially supported by the Simons
Foundation collaboration grant n. 691493

(To Yuri Ivanovich Manin, in memory.)

###### Abstract

In the present paper, dedicated to Yuri Manin, we investigate the general
notion of rings of ${{\mathbb S}}[\mu_{n,+}]$–polynomials and relate this
concept to the known notion of number systems. The Riemann-Roch theorem for
the ring ${\mathbb Z}$ of the integers that we obtained recently uses the
understanding of ${\mathbb Z}$ as a ring of polynomials ${{\mathbb S}}[X]$ in
one variable over the absolute base ${{\mathbb S}}$, where $1+1=X+X^{2}$. The
absolute base ${{\mathbb S}}$ (the categorical version of the sphere spectrum)
thus turns out to be a strong candidate for the incarnation of the mysterious
${\mathbb F}_{1}$.

##### Key Words.

Riemann-Roch, Number systems, Adeles, Zeta function, Sphere spectrum, Witt
vectors. .

##### Mathematics Subject Classification 2010:

##### Mathematics Subject Classification 2010:

14C40, 14G40, 14H05, 11R56, 13F35, 18G60, 19D55.

##  1 Introduction

> Les mathématiciens du xvi-ème siècle avaient coutume de parler de la
> “métaphysique du calcul infinitésimal”, de la “métaphysique de la théorie
> des équations”. Ils entendaient par là un ensemble d’analogies vagues,
> difficilement saisissables et difficilement formulables, qui néanmoins leur
> semblaient jouer un rôle important à un moment donné dans la recherche et la
> découverte mathématiques. (A. Weil, De la métaphysique aux mathématiques,
> 1960, [33])

Yuri Manin, to whose memory we dedicate this article, first recognized in [23]
the importance of developing a theory of “absolute coefficients” in arithmetic
geometry, independently of the early ideas proposed by R. Steinberg [30] and
J. Tits [31] in the context of Chevalley groups. In arithmetic, for number
fields, the goal is to provide the geometric counterpart to the construction
that A. Weil used in his proof of the Riemann hypothesis for function fields.
The search for a close analogy between number fields and function fields of
curves in positive characteristic induced Manin to postulate the existence of
the absolute point “${\rm Spec\,}{\mathbb F}_{1}$,” over which one could apply
Weil’s strategy to the study of the Riemann zeta function. For the algebraic
scheme ${\rm Spec\,}{\mathbb Z}$, one would then use the spectrum of the
tensor product “${\mathbb Z}\otimes_{{\mathbb F}_{1}}{\mathbb Z}$” as a
substitute for the self-product of a curve over (the spectrum of) a finite
field.
Manin always advocated the fruitfulness of unexpected interactions between
different approaches to a mathematical problem. In Sections 2 and 3 we shall
discuss two of such unexpected occurrences, in fact two pillars of our joint
work in the past fifteen years. Section 2 is about the hypothetical curve222we
reserve throughout the symbol $\bf C$ for this entity $\bf C$ that we propose
as the absolute geometric entity. Section 3 concerns instead the absolute
coefficients. The aim of this paper is to sponsor ${{\mathbb S}}$ the most
basic combinatorial form of the sphere spectrum and of an ${{\mathbb
S}}$-algebra, as the most natural candidate for the absolute coefficients (aka
${\mathbb F}_{1}$). We claim that this algebra is the absolute “field” of
constants over which ${\mathbb Z}$ becomes a ring of polynomials in one
variable. This point of view is supported by the Riemann-Roch theorem for the
ring ${\mathbb Z}$ recently proved in [14], whose formula shows that the genus
of ${\overline{{\rm Spec\,}{\mathbb Z}}}$ is zero. In an earlier result on the
same topic [13], the integers were considered as polynomials over ${{{\mathbb
S}}[\pm 1]}$ with generator $X=3$. This fact is based on the balanced ternary
numeral system333An early occurrence of this numeral system is found in the
1544 book “Arithmetica integra” of Michael Stifel. which provides a balanced
signed-digit representation of the integers as finite sums of powers of the
“variable” $X=3$ with coefficients in the set $\\{-1,0,1\\}$ underlying the
pointed multiplicative monoid $\mu_{2,+}$ of quadratic roots of unity. The new
version of the Riemann-Roch theorem for the ring ${\mathbb Z}$ in [14]
simplifies the earlier version [13] and it also reconciles the formula (and
our understanding of this subject) with the classical number theoretic
viewpoint. Indeed, in the analogy between number fields and curves over finite
fields, the field ${\mathbb Q}$ has genus zero [32] and it is singled out as
the only field contained in any other number field. The view of ${\mathbb Z}$
as a ring of polynomials over the absolute base ${{\mathbb S}}$ selects the
generator $X=-2$. The key fact is that any integer can be uniquely written as
a sum of powers of $-2$ [20].

The above special cases of generators $X$ for rings over finite spherical
${{\mathbb S}}$-algebras justify a systematic and broader study of rings of
${{\mathbb S}}[\mu_{n,+}]$–polynomials. In Section 5 we introduce the general
notion of rings of ${{\mathbb S}}[\mu_{n,+}]$–polynomials in one and several
variables. Let $n>0$ be an integer, $\mu_{n}$ the multiplicative group of
$n$-th roots of $1$ and ${{\mathbb S}}[\mu_{n,+}]$ the spherical ${{\mathbb
S}}$-algebra of the (pointed) monoid $\mu_{n,+}=\mu_{n}\cup\\{0\\}$. We recall
that morphisms of ${{\mathbb S}}$-algebras ${{\mathbb S}}[\mu_{n,+}]\to HR$
($R$ being a ring) correspond bijectively to group homomorphisms
$\iota:\mu_{n}\to R^{\times}$ [11]. Let ${\mathcal{P}}(\mu_{n})$ be the subset
of the set $\left(\mu_{n}\cup\\{0\\}\right)^{\mathbb N}$ of sequences with
only finitely many non-zero terms. By definition, an element $X\in R$ is an
${{\mathbb S}}[\mu_{n,+}]$-generator if and only if the evaluation map
$\sigma:{\mathcal{P}}(\mu_{n})\to R$,
$\sigma((\alpha_{j}))=\sum_{j}\iota(\alpha_{j})\,X^{j}$ is bijective.
Proposition 5.8 shows that the pair $(R,X)$ of a ring of ${{\mathbb
S}}[\mu_{n,+}]$–polynomials in one variable is uniquely specified, up to
isomorphism, by the map $h:\mu_{n}\to{\mathcal{P}}(\mu_{n})$, which, in turn,
is uniquely defined by the equality $\sigma(h(\xi))=\iota(\xi)+1$. In Section
6 we give several examples of rings of ${{\mathbb S}}[\mu_{n,+}]$–polynomials
based on some known number systems. We refer to [2] for a survey on systems of
numerations and for references therein contained, but we claim no
exhaustiveness. Conceptually, the examples of rings of ${{\mathbb
S}}[\mu_{n,+}]$-polynomials discussed in this article provide an explicit
bridge between the $p$-adic and the complex world. At the geometric level, the
rings of polynomials are naturally related to the projective line $\mathbb
P^{1}$, and the evaluation at the points $0$ and $\infty$ of $\mathbb P^{1}$
yields, after completion, the following refinement (the lower line) of a
classical diagram (upper line). In the upper line, $K$ is the field of
fractions of the $p$-typical Witt ring of the algebraic closure of ${\mathbb
F}_{q}$ ($q=p^{\ell}$) and $\overline{K}$ is its algebraic closure.

$\begin{array}[c]{ccccccccc}\overline{\mathbb
F}_{q}&\stackrel{{\scriptstyle\pi}}{{\twoheadleftarrow}}&W(\overline{\mathbb
F}_{q})&\hookrightarrow&\overline{K}&\supset&\overline{\mathbb
Q}&\subset&{\mathbb C}\\\
\rotatebox{90.0}{$\subset$}&&\rotatebox{90.0}{$\subset$}&&\rotatebox{90.0}{$\subset$}&&\rotatebox{90.0}{$\subset$}&&\rotatebox{90.0}{$=$}\\\
{\mathbb F}_{q}&\stackrel{{\scriptstyle\pi}}{{\twoheadleftarrow}}&W({\mathbb
F}_{q})&\hookrightarrow&W({\mathbb
F}_{q})[\eta]&\hookleftarrow&R[X^{-1}]&\hookrightarrow&{\mathbb C}\end{array}$

In the lower line, $X$ is a ${{\mathbb S}}[\mu_{n,+}]$-generator of the ring
$R$ where $n+1=q$. $R[X^{-1}]$ is the ring of Laurent polynomials; the map to
${\mathbb C}$ is the inclusion of $R[X^{-1}]$ in ${\mathbb C}$ by
specialization of $X$, obtained by solving the equations
$\sigma(h(\xi))=\iota(\xi)+1,\,\xi\in\mu_{n}$, and using the canonical
embedding $\mu_{n,+}\subset{\mathbb C}$. The map from $R[X^{-1}]$ to the
finite extension $W({\mathbb F}_{q})[\eta]$ is obtained from the canonical
inclusion of $R$ in the projective limit $\varprojlim R_{n}$ (see Proposition
5.8).

The general theory of rings of ${{\mathbb S}}[\mu_{n,+}]$-polynomials,
together with the role of the absolute base ${{\mathbb S}}$ in the formulation
of the Riemann-Roch theorem [14], suggest the following refinement of the
definition of the Arithmetic Site. Originally, this space was defined by the
pair of the arithmetic topos ${\widehat{{\mathbb N}^{\times}}}$ and the
structure sheaf given by the Frobenius action of ${\mathbb N}^{\times}$ on the
tropical semiring ${{\mathbb Z}_{\rm max}}$ [9]. The role of the field of
constants is here played by the Boolean semifield ${\mathbb B}$. The
development of this paper evidently hints to a replacement of the structure
sheaf ${{\mathbb Z}_{\rm max}}$ by the sheaf of ${{\mathbb S}}$-algebras
obtained from the Frobenius action $X\mapsto X^{n}$ of ${\mathbb N}^{\times}$
on the spherical algebra ${{\mathbb S}}[X]$. This new version of ${{\mathbb
S}}$-arithmetic site provides simultaneously a natural base both at the
coefficients and at the geometric levels. The topos ${\widehat{{\mathbb
N}^{\times}}}$ is the geometric incarnation of the $\lambda$-operations in the
theory of $\lambda$-rings [3] in the context of geometry over ${\mathbb
F}_{1}$. We expect that throughout a suitable understanding of the “algebraic
closure” $\overline{\mathbb F}_{1}$ of the absolute coefficients one may
relate the space of points of the ${{\mathbb S}}$-arithmetic site over
$\overline{\mathbb F}_{1}$ with the (points of the) curve $\bf C$ whose
structure is recalled in Section 2.

Finally, these results also point out to the open and interesting question of
the classification of rings of ${{\mathbb S}}[\mu_{n,+}]$–polynomials in
several variables which pursues the intuitive statement of Yuri Manin [23]:

> _The central question we address can be provocatively put as follows: if
> numbers are similar to polynomials in one variable over a finite field, what
> is the analog of polynomials in several variables? Or, in more geometric
> terms, does there exist a category in which one can define “absolute
> Descartes powers” ${\rm Spec\,}{\mathbb Z}\times\cdots\times{\rm
> Spec\,}{\mathbb Z}$?_

##  2 Adelic and topos theoretic incarnation of $\bf C$

A first connection between Manin’s point of view on ${\mathbb F}_{1}$ and a
seemingly unrelated topic takes place as a by-product of the relations between
C. Soulé perspective on varieties over ${\mathbb F}_{1}$ (named “Critical
Realism” in [24]) – motivated by Manin [23] (cf. §1.5) – and the work of the
first author [5] on the trace formula in noncommutative geometry and the zeros
of the Riemann zeta function. In [29], Soulé introduced the following zeta
function of a variety $X$ over ${\mathbb F}_{1}$

$\zeta_{X}(s):=\lim_{q\to 1}Z(X,q^{-s})(q-1)^{N(1)},\qquad s\in{\mathbb R}$
(2.1)

using the polynomial counting function $N(x)\in{\mathbb Z}[x]$ associated with
$X$ and the Hasse-Weil exponential series

$Z(X,T):=\exp\left(\sum_{r\geq 1}N(q^{r})\frac{T^{r}}{r}\right).$ (2.2)

All the examples of varieties considered in op.cit. are rational. Thus, the
existence of an underlying curve $\bf C$ related, in a similar manner, to the
Riemann zeta function is subordinated to finding a function $N(q)$ (highly
non-polynomial!) that produces, through the use of (2.1), the complete Riemann
zeta function $\zeta_{\mathbb Q}(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$. This is a
non-trivial problem since classically, $N(1)$ in the above formula inputs the
Euler characteristic of the geometric space. Thus one might be induced to
expect444the number of zeros of $\zeta_{\mathbb Q}$ is infinite, and so is the
dimension of the (mysterious) cohomology $H^{1}(\bf C)$ that since for the
Riemann zeta-function one ought to have $N(1)=-\infty$, the use of (2.1)
should be precluded, and with it also the expectation that $N(q)\geq 0$ for
$q\in(1,\infty)$. There is, in fact, a natural way to by-pass this problem by
applying the logarithmic derivatives to both sides of (2.1) and then observing
that the right-hand side determines the Riemann sums of an integral [7, 8]. In
this way, in place of (2.1) one considers the equation:
$\frac{\partial_{s}\zeta_{N}(s)}{\zeta_{N}(s)}=-\int_{1}^{\infty}N(u)\,u^{-s}d^{*}u,$
where $d^{*}u:=du/u$. This integral formula produces the following one for the
sought for counting function $N(q)$ associated with $\bf C$:

$\frac{\partial_{s}\zeta_{\mathbb Q}(s)}{\zeta_{\mathbb
Q}(s)}=-\int_{1}^{\infty}N(u)\,u^{-s}d^{*}u.$ (2.3)

The above equation admits a meaningful solution expressable in terms of the
distribution

$N(u)=\frac{d}{du}\varphi(u)+\kappa(u),\qquad\varphi(u):=\sum_{n<u}n\,\Lambda(n),$
(2.4)

where $\kappa(u)$ is the distribution that appears in the Riemann-Weil
explicit formula

$\int_{1}^{\infty}\kappa(u)f(u)d^{*}u=\int_{1}^{\infty}\frac{u^{2}f(u)-f(1)}{u^{2}-1}d^{*}u+cf(1)\,,\qquad
c=\frac{1}{2}(\log\pi+\gamma).$

One shows that the distribution $N(u)$ is positive on $(1,\infty)$, and when
written in terms of the non-trivial zeros $\rho\in Z$ of the Riemann zeta
function, it is given, in complete analogy with its counterpart holding in the
function field case, by

$N(u)=u-\frac{d}{du}\left(\sum_{\rho\in Z}{\rm
order}(\rho)\frac{u^{\rho+1}}{\rho+1}\right)+1,$ (2.5)

where the derivative is taken in the sense of distributions. The value at
$u=1$ of the term $\displaystyle{\omega(u)=\sum_{\rho\in Z}{\rm
order}(\rho)\frac{u^{\rho+1}}{\rho+1}}$ is given by
$\frac{1}{2}+\frac{\gamma}{2}+\frac{\log
4\pi}{2}-\frac{\zeta^{\prime}(-1)}{\zeta(-1)}$.

Figure 1: Graph of a primitive $J(u)$ of the counting distribution $N(u)$. One
has $J(u)\to-\infty$ when $u\to 1$. The wiggly graph is the approximation of
$J(u)$ obtained using the symmetric set $Z_{m}$ of the first $2m$ zeros to
perform the sum $J_{m}(u)=\frac{u^{2}}{2}-\sum_{Z_{m}}{\rm
order}(\rho)\frac{u^{\rho+1}}{\rho+1}+u$.

The tension between the positivity of the distribution $N(q)$ for $q>1$, and
the expectation that its value $N(1)$ ought to be $N(1)=-\infty$ is resolved
by implementing the theory of distributions. Indeed, even though $N(u)$ is
finite as a distribution, when one looks at it as a function, its value at
$q=1$ is formally given by

$N(1)=2-\lim_{\epsilon\to
0}\frac{\omega(1+\epsilon)-\omega(1)}{\epsilon}\sim-\frac{1}{2}E\log E,\qquad\
E=\frac{1}{\epsilon}$

thus, it is $-\infty$, and this fact reflects, when $\epsilon\to 0$, the
density of the zeros of the zeta function.

We emphasize that the role of the Riemann-Weil explicit analytic formulas in
the process of overcoming the initial difficulty through a solution defined by
a positive distribution $N(q)$, directly connects the original (classical
geometric) viewpoint with the trace formula in [5], thus providing a first
geometric description for the points of $\bf C$ in terms of the double
quotient

$X_{\mathbb Q}:={\mathbb Q}^{\times}\backslash{\mathbb A}_{\mathbb
Q}/{\hat{\mathbb Z}^{\times}}$ (2.6)

of the adele class space of the rationals by the maximal compact subgroup
${\hat{\mathbb Z}^{\times}}$ of the idele class group. The main key player in
this construction is the scaling action of ${\mathbb R}_{+}^{\times}$ which
provides555To remove the divergent logarithmic term from the trace formula [5]
one needs to remove from $X_{\mathbb Q}$ the orbit of the unit adele $1$, i.e.
equivalently to subtract the regular representation of ${\mathbb
R}_{+}^{\times}$ as in [25]. the above counting distribution $N(u)$,
$u\in[1,\infty)$, that determines, in turn, the complete Riemann zeta function
via a limiting procedure as $q\to 1$, operated on the Hasse-Weil formula.
Noncommutative geometry plays a crucial role in this development mainly by
implementing the noncommutative space $X_{\mathbb Q}$ which naturally arises
as the dual of the BC-system [4].

To achieve a more classical geometric understanding of the adele class space
$X_{\mathbb Q}$ with its scaling action, in analogy with the action of the
Frobenius automorphism on the points of a curve over the algebraic closure of
a ground field, one needs to push further the search of other unexpected
interactions… This geometric understanding comes in fact from the interplay
among three a priori unrelated theories

  1. 1.

Noncommutative Geometry

  2. 2.

Grothendieck topoi

  3. 3.

Tropical Geometry.

The natural starting point is the topos ${\widehat{{\mathbb N}^{\times}}}$,
defined in [9] as the Grothendieck presheaf topos dual to the multiplicative
monoid ${\mathbb N}^{\times}$ of non-zero positive integers. This space is in
fact the geometric incarnation of ${\mathbb N}^{\times}$-actions on sets.
These actions often occur in global instances of Frobenius endomorphisms: for
$\lambda$-rings they were advocated in [3] in the context of varieties over
${\mathbb F}_{1}$ (”Futurism” in Manin’s interpretation, [24]). Special
$\lambda$-rings $R$ ([1] Proposition 5.2), belong naturally to the topos
${\widehat{{\mathbb N}^{\times}}}$ since the Adams operations $\psi_{n}$ turn
$R$ into a sheaf of rings on the topos ${\widehat{{\mathbb N}^{\times}}}$.
At a very basic algebraic level, a fundamental example of Frobenius action of
${\mathbb N}^{\times}$ occurs in the theory of semirings (i.e. when one drops
the existence of the additive inverse in rings). For a semifield666A semifield
is a semiring whose non-zero elements form a group under multiplication $R$ of
“characteristic one” (aka idempotent: i.e. such that $1+1=1$), the map
$x\mapsto x^{n}={\rm Fr}_{n}(x)$ is an injective endomorphism [17], for any
integer $n\in{\mathbb N}^{\times}$. Thus, one obtains a canonical action of
the semigroup ${\mathbb N}^{\times}$ on any such $R$. For this reason it is
natural to work with the topos ${\widehat{{\mathbb N}^{\times}}}$ endowed with
an action of ${\mathbb N}^{\times}$. Furthermore, one also knows that there is
a unique semifield777As a multiplicative monoid ${{\mathbb Z}_{\rm max}}$ is
obtained by adjoining the zero element $-\infty$ to the infinite cyclic group
${\mathbb Z}$ while the operation which plays the role of addition in the
semifield is $(x,y)\mapsto\max(x,y)$ ${{\mathbb Z}_{\rm max}}$ whose
multiplicative group is infinite cyclic and it is of characteristic one. Given
these facts, it is natural to introduce the following space

######  Definition 2.7 ([9]).

The Arithmetic Site ${\mathscr{A}}={({\widehat{{\mathbb
N}^{\times}}},{\mathcal{O}})}$ is the topos ${\widehat{{\mathbb N}^{\times}}}$
endowed with the structure sheaf ${\mathcal{O}}:={{\mathbb Z}_{\rm max}}$,
viewed as a semiring in the topos and with the action of ${\mathbb
N}^{\times}$ by Frobenius endomorphisms.

The semifield ${{\mathbb Z}_{\rm max}}$ and its companion ${\mathbb
R}_{+}^{\rm max}$ (whose multiplicative group is ${\mathbb R}_{+}^{*}$), are
familiar objects in tropical geometry where the maximum substitutes the usual
addition.
By implementing a straightforward generalization in semi-ringed toposes of the
understanding of a point in algebraic geometry, one obtains the following
result which determines a bridge connecting noncommutative geometry with
(Grothendieck) topos theory

######  Theorem 2.8 ([9]).

The set of points of the arithmetic site ${\mathscr{A}}$ over ${\mathbb
R}_{+}^{\rm max}$ is canonically isomorphic to $X_{\mathbb Q}={\mathbb
Q}^{\times}\backslash{\mathbb A}_{\mathbb Q}/{\hat{\mathbb Z}^{\times}}$. The
action of the Frobenius automorphisms ${\rm Fr}_{\lambda}$ of ${\mathbb
R}_{+}^{\rm max}$ on these points corresponds to the action of the idele class
group on $X_{\mathbb Q}={\mathbb Q}^{\times}\backslash{\mathbb A}_{\mathbb
Q}/{\hat{\mathbb Z}^{\times}}$.

This theorem sheds new light on a geometric intuition of the curve $\bf C$, in
particular, it displays the noncommutative space $X_{\mathbb Q}$ as the set of
points of $\bf C$ over the semifield ${\mathbb R}_{+}^{\rm max}$, with the
scaling action understood as the action of the Galois group ${\rm
Aut}_{\mathbb B}({\mathbb R}_{+}^{\rm max})$ of ${\mathbb R}_{+}^{\rm max}$
over the Boolean semifield888${\mathbb B}:=\\{0,1\\}$ with $1+1=1$ ${\mathbb
B}$. It also suggests that ${\mathbb R}_{+}^{\rm max}$ ought to be involved in
the construction of the “algebraic closure” of ${\mathbb F}_{1}$, and that the
combinatorial core underlying $\bf C$ is countable since both ${\mathbb
N}^{\times}$ and ${{\mathbb Z}_{\rm max}}$ are so. We find quite remarkable
that while the Arithmetic Site is a combinatorial object of countable nature,
it comes nonetheless endowed with a one-parameter semigroup of
“correspondences” which can be viewed as congruences on the square of this
site [9].

The countable set of places of ${\mathbb Q}$ (the points of the Arakelov
compactification ${\overline{{\rm Spec\,}{\mathbb Z}}}$), is the (classically)
visible analog of the set of the orbits of the Frobenius automorphism in the
function field case. One obtains a better view of the points of $\bf C$ by
considering the periodic orbits $C_{p}$ (parameterized by primes $p$) as they
occur among the points of the Arithmetic Site ${\mathscr{A}}$ over ${\mathbb
R}_{+}^{\rm max}$. One shows that the points of $C_{p}$ form a circle whose
elements are rank-one subgroups of the multiplicative group of ${\mathbb
R}_{+}^{\rm max}$ of the form

$H_{\mu}:=\\{\mu^{\frac{n}{p^{k}}}\mid n\in{\mathbb Z},\ k\in{\mathbb N}\\}.$
(2.9)

This subgroup is unchanged if one replaces $\mu$ with $\mu^{p}$, and the
Frobenius action of ${\rm Aut}_{\mathbb B}({\mathbb R}_{+}^{\rm max})={\mathbb
R}_{+}^{*}$, $\mu\mapsto\mu^{\lambda}$, induces the transitive action of the
quotient group ${\mathbb R}_{+}^{*}/p^{\mathbb Z}$. The length of this
periodic orbit is $\log p$, and their full collection plays a key role in the
trace formula interpretation of the Riemann-Weil explicit formulas in [5].
Moreover, each $C_{p}$ inherits, as a subspace of the Scaling Site (obtained
from the Arithmetic Site by extension of scalars), a structure sheaf (of
characteristic one) which turns each periodic orbit into the analog of a
classical elliptic curve [10]. In this way, one can still apply several key
tools of algebraic geometry, such as rational functions, divisors, etc. A
striking new feature of the geometry of a periodic orbit is that the degree of
a divisor is a real number. For any divisor $D$ in $C_{p}$, there is a
corresponding Riemann-Roch problem with solution space $H^{0}(D)$. The
continuous dimension999In analogy with von-Neumann’s continuous dimensions of
the theory of type II factors ${{\mbox{Dim}_{\mathbb R}}}(H^{0}(D))$ of this
${\mathbb R}_{+}^{\rm max}$-module is defined by the limit

${{\mbox{Dim}_{\mathbb
R}}}(H^{0}(D)):=\lim_{n\to\infty}p^{-n}{{\mbox{dim}_{\rm
top}}}(H^{0}(D)^{p^{n}})$ (2.10)

where $H^{0}(D)^{p^{n}}$ is a naturally defined filtration and
${{\mbox{dim}_{\rm top}}}({\mathcal{E}})$ denotes the topological dimension of
an ${\mathbb R}_{+}^{\rm max}$-module ${\mathcal{E}}$. The following Riemann-
Roch formula holds

######  Theorem 2.11 ([10]).

$(i)$ Let $D\in{\rm Div}(C_{p})$ be a divisor with $\deg(D)\geq 0$. Then the
limit in (2.10) converges and one has

${{\mbox{Dim}_{\mathbb R}}}(H^{0}(D))=\deg(D).$

$(ii)$ The following Riemann-Roch formula holds

${{\mbox{Dim}_{\mathbb R}}}(H^{0}(D))-{{\mbox{Dim}_{\mathbb
R}}}(H^{0}(-D))=\deg(D)\qquad\forall D\in{\rm Div}(C_{p}).$

In view of these results and the leading role played by the Boolean semifield
${\mathbb B}$ among algebraic idempotent structures101010${\mathbb B}$ is, in
particular, the only finite semifield that is not a field cf. [17], one might
be (wrongly) induced to think of ${\mathbb B}$ as the natural incarnation of
${\mathbb F}_{1}$. However, this cannot be the case for the straightforward
reason that111111algebras over ${\mathbb B}$ are of characteristic one:

The ring ${\mathbb Z}$ is not an algebra over ${\mathbb B}$.

##  3 The absolute coefficients, spectra and ${{\mathbb S}}$.

The above undeniable fact led us, once again, to compare Manin’s ideas on
${\mathbb F}_{1}$ with another a priori unrelated topic: this is the world of
homotopy theory spectra. Topological spectra greatly generalize cohomology
theories; many important invariants in algebraic topology, like ordinary
cohomology and K-theory, can be reformulated in terms of spectra, which thus
provide a unified treatment for “generalized coefficients”. One fundamental
discovery in the topological context is that “ring spectra” generalize rings,
and in particular, the “sphere spectrum” $\underline{{{\mathbb S}}}$ becomes
more basic than the ring ${\mathbb Z}$, because the latter can be seen as an
algebra over the former. This theory of “brave new rings” has proved to be the
right framework for cyclic homology; in particular, the theory of
$\Gamma$-spaces is known to provide a workable model of connective spectra
[15]. One usually works at the homotopy level, so it is crucial to handle Kan
complexes to obtain a good model structure. However, to take full advantage of
this theory for the development of Manin’s ideas on ${\mathbb F}_{1}$ in
number theory, we believe that $\Gamma$-spaces ought to be viewed in their
most basic form, namely as simplicial objects in the category of
$\Gamma$-sets, so that homotopy theory can play the role of homological
algebra corresponding to the “absolute algebra” over the base $\Gamma$-ring
${{\mathbb S}}$ [11]. This $\Gamma$-ring is the categorical starting point in
the construction of the sphere spectrum $\underline{{{\mathbb S}}}$, together
with the natural functor from $\Gamma$-spaces to spectra, and its is exactly
this basic combinatorial nature that makes it closer to the sought for
${\mathbb F}_{1}$. The category ${\Gamma{\mathfrak{Sets}_{*}}}$ of pointed
$\Gamma$-sets (aka ${{\mathbb S}}$-modules ${\mathfrak{Mod}}({{\mathbb S}})$)
can be directly described as follows. One starts with the small category
$\Gamma^{\rm op}$ as a full subcategory of the category of finite pointed sets
whose objects are the pointed finite sets121212where $0$ is the base point.
$k_{+}:=\\{0,\ldots,k\\}$, for $k\geq 0$. In particular, $0_{+}$ is both
initial and final in $\Gamma^{\rm op}$, making ${\Gamma^{\rm op}}$ a pointed
category. A $\Gamma$-set is defined as a (covariant) functor ${\Gamma^{\rm
op}}\longrightarrow{\mathfrak{Sets}_{*}}$ between pointed categories, and the
morphisms in this category are natural transformations. One lets ${{\mathbb
S}}:{\Gamma^{\rm op}}\longrightarrow{\mathfrak{Sets}_{*}}$ be the inclusion
functor. The internal hom functor is defined by

$\underline{{\rm{Hom}}}_{{\mathbb
S}}(M,N):=\\{k_{+}\mapsto{\rm{Hom}}_{{\mathbb S}}(M,N(k_{+}\wedge-))\\}.$

This formula uniquely defines the smash product of $\Gamma$-sets by applying
the adjunction

$\underline{{\rm{Hom}}}_{{\mathbb S}}(M_{1}\wedge
M_{2},N)=\underline{{\rm{Hom}}}_{{\mathbb
S}}(M_{1},\underline{{\rm{Hom}}}_{{\mathbb S}}(M_{2},N)).$

The basic construction of ${{\mathbb S}}$-modules associates to an abelian
monoid $A$ with a zero element, the Eilenberg-MacLane functor $M=HA$

$HA(k_{+})=A^{k},\qquad Hf:HA(k_{+})\to HA(n_{+}),$ $\
Hf(m)(j):=\sum_{f(\ell)=j}m_{\ell},$

where $m=(m_{1},\ldots,m_{k})\in HA(k_{+})$, and the zero element of $A$ gives
meaning to the empty sum. An ${{\mathbb S}}$-algebra $\mathcal{A}$ is an
${{\mathbb S}}$-module $\mathcal{A}:{\Gamma^{\rm
op}}\longrightarrow{\mathfrak{Sets}_{*}}$ endowed with an associative
multiplication $\mu:{\mathcal{A}}\wedge{\mathcal{A}}\to{\mathcal{A}}$ and a
unit $1:{{\mathbb S}}\to{\mathcal{A}}$.
An ordinary semiring $R$ gives rise to the ${{\mathbb S}}$-algebra $HR$, and
the corresponding embedding of categories is fully faithful so that no
information is lost. In contrast, the basic ${{\mathbb S}}$-algebra ${{\mathbb
S}}$ now lies under $HR$ for any semiring $R$.
Given a multiplicative monoid $M$ with a zero element $0\in M$ such that
$0\times x=x\times 0=0$ for all $x\in M$, one defines the spherical ${{\mathbb
S}}$-algebra ${{\mathbb S}}[M]$ which associates to the pointed set $X$ the
smash product $X\wedge M$, where the base point of $M$ is $0\in M$. One
identifies ${{\mathbb S}}[M][1_{+}]=1_{+}\wedge M$ with $M$ by sending the
base point of $1_{+}\wedge M$ to $0\in M$, and $a\wedge m$ where $a\in
1_{+}\setminus\\{*\\}$ and $m\in M\setminus\\{0\\}$ to $m$. To avoid confusion
we write $2_{+}=\\{*,a,b\\}$. Besides the base point the elements of
${{\mathbb S}}[M][2_{+}]=2_{+}\wedge M$ are given by pairs of the form $(a,m)$
or $(b,m)$ where $m\in M\setminus\\{0\\}$. One has three natural pointed maps
$f:2_{+}\to 1_{+}$, which are

$\phi(a)=a,\ \phi(b)=*,\ \ \psi(a)=*,\ \psi(b)=a,\ \ \rho(a)=\rho(b)=a.$

Let $m\in M\setminus\\{0\\}$ and consider the pair $z=(b,m)\in{{\mathbb
S}}[M][2_{+}]$. One has $\phi_{*}(z)=*=0$ and $\psi_{*}(z)=m$. Moreover one
has $\rho_{*}(z)=m$. This means that for the partially defined addition in
${{\mathbb S}}[M][1_{+}]=M$, one has $0+m=m$ for all $m\in M$.
Thus both ordinary rings and monoids fit fully faithfully and naturally [11]
(Proposition 3.5), in the category of ${{\mathbb S}}$-algebras yielding a
strong argument for viewing ${{\mathbb S}}$ as the natural candidate for
${\mathbb F}_{1}$. Nonetheless one needs to test this idea in various ways.
For instance, one sees op.cit. that the tensor square of $H{\mathbb Z}$ over
${{\mathbb S}}$ is non-isomorphic to $H{\mathbb Z}$, and this result provides
more ground to the original intuition of Manin in [23]. One may also wonder
which advancements this point of view may produce to the understanding of the
ring ${\mathbb Z}$ and its algebraic spectrum ${\rm Spec\,}{\mathbb Z}$. We
shall now move to a detailed discussion of this topic.

Let ${\overline{{\rm Spec\,}{\mathbb Z}}}$ be the Arakelov compactification of
${\rm Spec\,}{\mathbb Z}$ obtained by adding the archimedean place with
associated symbol $\infty$. Then, the new point of view described above
provides a natural extension of the classical structure sheaf of ${\rm
Spec\,}{\mathbb Z}$ to the Arakelov compactification. The crucial points
concerning the quest for the curve $\bf C$ are two: firstly, this extended
structure sheaf ${\mathcal{O}}$ is still a subsheaf of the constant sheaf
${\mathbb Q}$; the second interesting point is that the global sections of
${\mathcal{O}}$ form a finite algebra extension of ${{\mathbb S}}$. This
extension is identifiable with the extension by the two roots of unity inside
${\mathbb Q}$ that we used in [6] in the process of showing that Chevalley
groups are varieties over ${\mathbb F}_{1^{2}}$ in the sense of
Soulé131313Another convincing argument in favor of ${{\mathbb S}}$-algebras is
that the ad-hoc category we introduced in [7] to simplify Soulé’s definition
of varieties over ${\mathbb F}_{1}$, is naturally (see [12]) a full
subcategory of the category of ${{\mathbb S}}$-algebras. The condition that
restricts the elements of $H{\mathbb Q}$ at the archimedean place is simple to
formulate when one views the functor $H{\mathbb Q}$ as assigning to a finite
pointed set $F$ the ${\mathbb Q}$-valued divisors on $F$. The restriction is
then stated by writing that the sum of the absolute values of the involved
rational numbers is $\leq 1$. One checks that this condition is stable under
push-forwards and products and hence it defines a sub-${{\mathbb S}}$-algebra
of $H{\mathbb Q}$. This sub-${{\mathbb S}}$-algebras, defined using a norm,
also applies in the context of the adeles of a global field and allows one to
transpose the approach due to A. Weil of the Riemann-Roch theorem for function
fields to the number field ${\mathbb Q}$ [13].

A divisor $D$ on ${\overline{{\rm Spec\,}{\mathbb Z}}}$ defines a compact
subset $K=\prod K_{v}\subset{\mathbb A}_{\mathbb Q}$ of the locally compact
ring of adeles. When $p$ is a non-archimedean prime, each
$K_{p}\subset{\mathbb Q}_{p}$ is an additive subgroup, in contrast, the
compact subset $K_{\infty}\subset{\mathbb R}$ is just a symmetric interval
whose lack of additive structure prevents one to use Weil’s original
construction involving the addition map $\psi:{\mathbb Q}\times K\to{\mathbb
A}_{\mathbb Q}$. On the other hand, one also quickly notices that $\psi$
retains its meaning in the context of ${{\mathbb S}}$-modules, giving rise to
a short complex. Using the Dold-Kan correspondence in the context of
${{\mathbb S}}$-algebras, one then introduces a $\Gamma$-space $H(D)$ which
encodes the homological information of the divisor $D$ and only depends upon
the linear equivalence class of $D$ (i.e. the divisor class is unchanged under
the multiplicative action of ${\mathbb Q}^{\times}$ on ${\mathbb A}_{\mathbb
Q}$). As a by-product, one obtains a Riemann-Roch formula for Arakelov
divisors on ${\overline{{\rm Spec\,}{\mathbb Z}}}$ of an entirely novel nature
that relies on the introduction of three new key notions: (integer) dimension,
cohomologies $(H^{0}(D),H^{1}(D))$ (attached to a divisor $D$), and Serre
duality. More precisely, the Riemann-Roch formula equates the integer-valued
Euler characteristic with a simple modification of the traditional expression
(i.e. the degree of the divisor plus log 2).

######  Theorem 3.1 ([13]).

Let $D$ be an Arakelov divisor on ${\overline{{\rm Spec\,}{\mathbb Z}}}$. Then

$\dim_{{{{\mathbb S}}[\pm 1]}}H^{0}(D)-\dim_{{{{\mathbb S}}[\pm
1]}}H^{1}(D)=\bigg{\lceil}\deg_{3}D+\log_{3}2\bigg{\rceil}-\mathbf{1}_{L}.$
(3.2)

Here, $\lceil x\rceil$ denotes the odd function on ${\mathbb R}$ that agrees
with the ceiling function on positive reals, and $\mathbf{1}_{L}$ is the
characteristic function of an exceptional set141414$L\subset{\mathbb R}$ is
the union, for $k\geq 0$, of the intervals
$\deg(D)\in(\log\frac{3^{k}}{2},\log\frac{3^{k}+1}{2})$ of finite Lebesgue
measure.

In (3.2), the neperian logarithm that is traditionally used to define the
degree of a divisor $D=\sum_{j}a_{j}\\{p_{j}\\}+a\\{\infty\\}$ in Arakelov
geometry, is replaced by the logarithm in base $3$. This alteration is
equivalent to the division by $\log 3$ i.e. $\deg_{3}(D):=\deg(D)/\log 3$,
$\log_{3}2=\log 2/\log 3$.
The number $3$ appears unexpectedly in the computation of the dimension of the
cohomology of the ${{{\mathbb S}}[\pm 1]}$-modules by determining their
minimal number of linear generators. For $\dim_{{{{\mathbb S}}[\pm
1]}}H^{0}(D)$ one finds that the most economical way of writing the elements
of a symmetric interval ${\mathbb Z}\cap K_{\infty}$ involves writing integers
as polynomials of the form

$\sum_{j\geq 0}\alpha_{j}\ 3^{j},\ \ \alpha_{j}\in\\{-1,0,1\\}.$ (3.3)

Similarly, in the case of $\dim_{{{{\mathbb S}}[\pm 1]}}H^{1}(D)$, one finds
that the best way to approximate elements of the circle ${\mathbb R}/{\mathbb
Z}$ is to use Laurent polynomials of the form

$\sum_{j<0}\alpha_{j}\ 3^{j},\ \ \alpha_{j}\in\\{-1,0,1\\}.$ (3.4)

The key fact here is that the addition151515once the addition is defined, the
product follows uniquely using $X^{j}\,X^{k}=X^{j+k}$ of polynomials
$P(X)=\sum_{j\geq 0}\alpha_{j}\ X^{j},\ \ \alpha_{j}\in\\{-1,0,1\\}$ with
coefficients in ${{{\mathbb S}}[\pm 1]}$ is identical to the addition of
(truncated) Witt vectors for the finite field ${\mathbb F}_{3}$. One finds
that the addition $P+Q$ of two polynomials of degree $\leq n$ gives a
polynomial of degree $\leq n+1$, and that the only non-obvious rule one has to
prescribe is the sum: $1+1:=X-1$. Conceptually, the fundamental point is that
the image of the Teichmuller lift for ${\mathbb F}_{3}$ sits inside ${\mathbb
Z}$. At the same time, the Witt vectors with only finitely many non-zero
components form a subring of the Witt ring, and this subring is ${\mathbb Z}$!

##  4 The ring of integers as a ring of polynomials

There is another way to represent the integers as polynomials in one variable,
and in this description, the “coefficients” belong to the absolute base
${{\mathbb S}}$. This representation is known as the negabinary representation
of numbers

$n=\sum\alpha_{j}\ (-2)^{j},\ \ \alpha_{j}\in\\{0,1\\}.$ (4.1)

The number $X=-2$ is remarkably unique, making the representation of an
integer $n$ possible as polynomial $P(X)$ with coefficients
$\alpha_{j}\in\\{0,1\\}$. By following the same steps that led us to Theorem
3.1, but working now over the absolute base ${{\mathbb S}}$, one obtains the
following new and simplified version of the Riemann-Roch formula which now
involves the logarithm in base $2$

######  Theorem 4.2 ([14]).

Let $D$ be an Arakelov divisor on ${\overline{{\rm Spec\,}{\mathbb Z}}}$. Then

$\dim_{{{\mathbb S}}}H^{0}(D)-\dim_{{{\mathbb
S}}}H^{1}(D)=\bigg{\lceil}\deg_{2}D\bigg{\rceil}^{\prime}+1$ (4.3)

where $\lceil x\rceil^{\prime}$ is the right continuous function which agrees
with ceiling$(x)$ for $x>0$ non-integer and with $-$ceiling$(-x)$ for $x<0$
non-integer.

This version of the Riemann-Roch Theorem improves on Theorem 3.1 for the
following reasons:

  1. 1.

The term $\mathbf{1}_{L}$ involving the exceptional set $L$ in the original
statement (see [13]) has now disappeared from the formula.

  2. 2.

The formula (4.3) is in perfect analogy with the Riemann-Roch theorem for
curves of genus $0$.

  3. 3.

The canonical divisor $K=-2\\{2\\}$ has integral degree $\deg_{2}(K)=-2$.

Theorem 4.2 fits now perfectly with the tri-lingual text suggested by A. Weil,
that supports the analogy between Riemann’s transcendental theory of algebraic
functions of one variable in the first column, the algebraic geometry of
curves over finite fields, in the middle column, and the theory of algebraic
number fields in the third column. Indeed, according to Weil

> _Mais on peut, je crois, en donner une idée imagée en disant que le
> mathématicien qui étudie ces problèmes, a l’impression de déchiffrer une
> inscription trilingue. Dans la première colonne se trouve la théorie
> riemannienne des fonctions algébriques au sens classique. La troisième
> colonne c’est la théorie arithmétique des nombres algébriques. La colonne du
> milieu est celle dont la découverte est la plus récente : elle contient la
> théorie des fonctions algébriques sur un corps de Galois. Ces textes sont
> l’unique source de nos connaissances sur les langues dans lesquels ils sont
> écrits; de chaque colonne, nous n’avons bien entendu que des fragments ; la
> plus complète et celle que nous lisons le mieux, encore à présent, c’est la
> première. Nous savons qu’il y a de grandes différences de sens d’une colonne
> à l’autre, mais rien ne nous en avertit à l’avance. Á l’usage, on se fait
> des bouts de dictionnaire, qui permettent de passer assez souvent d’une
> colonne à la colonne voisine._

In Weil’s vision there is, in the middle column (that of function fields), a
geometric understanding of the zeta function as the generating function of the
number of points of the curve over extensions of the field of constants. In
section 2 we translated in this dictionary the Hasse-Weil formula, thus
leading one to the first encounter with the “the curve” $\bf C$ and the action
of ${\mathbb R}^{*}_{+}$ on $\bf C$, analogous to a Galois action. Theorem 4.2
indicates that the role of the field of constants is played by the absolute
coefficient ring ${{\mathbb S}}$. Since the boolean semifield ${\mathbb B}$
can be viewed as a ${{\mathbb S}}$-algebra, this translation suggests to
descend the structures of the Arithmetic and Scaling Sites discussed in
section 2 from ${\mathbb B}$ to ${{\mathbb S}}$.

##  5 Rings of ${{\mathbb S}}[\mu_{n,+}]$-polynomials

Let $n>0$ be an integer, $\mu_{n}$ the multiplicative group of $n$-th roots of
$1$ and ${{\mathbb S}}[\mu_{n,+}]$ the spherical ${{\mathbb S}}$-algebra of
the (pointed) monoid $\mu_{n,+}=\mu_{n}\cup\\{0\\}$. We recall that morphisms
of ${{\mathbb S}}$-algebras ${{\mathbb S}}[\mu_{n,+}]\to HR$ correspond
(bijectively) to group homomorphisms $\iota:\mu_{n}\to R^{\times}$ [11]. In
this section, we introduce the notion of rings of ${{\mathbb
S}}[\mu_{n,+}]$-polynomials in one (Definition 5.1) and several variables
(Remark 5.2) which might play a key role in the search of the “absolute
Descartes powers” among ordinary rings. We show that the pair $(R,X)$ of a
ring $R$ and an ${{\mathbb S}}[\mu_{n,+}]$-generator of $R$ is uniquely
characterized, up to isomorphism, by the map from $\mu_{n}$ to polynomials
with coefficients in the pointed monoid $\mu_{n,+}$, that encodes the addition
of $1$ into elements of $\mu_{n}$.

######  Definition 5.1.

Let $R$ be a ring, $\iota:\mu_{n}\to R^{\times}$ be an injective group
homomorphism. An element $X\in R$ is an ${{\mathbb S}}[\mu_{n,+}]$-generator
of $R$ if and only if every element $z\in R$ can be written uniquely as a
polynomial $z=\sum_{j}\iota(\alpha_{j})\,X^{j}$ with coefficients
$\alpha_{j}\in\mu_{n}\cup\\{0\\}$.

######  Remark 5.2.

More generally, a finite set $\\{X_{i}\mid i\in\\{1,\ldots,k\\}\\}$,
${{\mathbb S}}[\mu_{n,+}]$-generates $R$ if and only if every element $z\in R$
can be written uniquely as a polynomial $z=\sum_{j}\iota(\alpha_{j})\,X^{j}$
with coefficients $\alpha_{j}\in\mu_{n}\cup\\{0\\}$, where $j$ is a multi-
index $j=(j_{1},\ldots,j_{k})\in{\mathbb N}^{k}$, and $X^{j}:=\prod
X_{i}^{j_{i}}$.

Let ${\mathcal{P}}(\mu_{n})$ be the subset of the set
$\left(\mu_{n}\cup\\{0\\}\right)^{\mathbb N}$ of sequences with only finitely
many non-zero terms. Let $X\in R$, then the map
$\sigma:{\mathcal{P}}(\mu_{n})\to R$, given by

$\sigma((\alpha_{j})):=\sum_{j}\iota(\alpha_{j})\,X^{j}$ (5.3)

is well defined since for $\alpha=(\alpha_{j})\in{\mathcal{P}}(\mu_{n})$ the
sum $\sum_{j}\iota(\alpha_{j})\,X^{j}$ defines an element of $R$. It follows
from Definition 5.1 that if $X$ is an ${{\mathbb S}}[\mu_{n,+}]$-generator,
the map $\sigma$ is a bijection of ${\mathcal{P}}(\mu_{n})$ with $R$.

The simplest instance of a ${{\mathbb S}}[\mu_{n,+}]$ generator, with $n+1$ a
prime power $q$, is provided by the following example.

######  Example 5.4.

The ring $R={\mathbb F}_{q}[X]$ of polynomials over the finite field ${\mathbb
F}_{q}$ has the variable $X$ as ${\mathbb F}_{q}^{\times}$-generator.

Next proposition shows that the $m$-th root of an ${{\mathbb
S}}[\mu_{n,+}]$-generator $X$ of a ring $R$ is a ${{\mathbb
S}}[\mu_{n,+}]$-generator of the $R$-algebra extension $R[Y]/(Y^{m}-X)$, hence
providing an infinite source of examples.

######  Proposition 5.5.

Let $R$ be a ring, $\iota:\mu_{n}\to R^{\times}$ be an injective group
homomorphism, $X\in R$ an ${{\mathbb S}}[\mu_{n,+}]$-generator of $R$ and
$m\in{\mathbb N}$ be a positive integer. Then $Y\in R[Y]/(Y^{m}-X)$ is an
${{\mathbb S}}[\mu_{n,+}]$-generator of $R[Y]/(Y^{m}-X)$.

###### Proof.

Any element $z$ of $R[Y]/(Y^{m}-X)$ can be written uniquely as
$z=\sum_{j=0}^{m-1}a_{j}Y^{j}$, with $a_{j}\in R$ written uniquely as
$a_{j}=\sum_{j,k}\iota(\alpha_{j,k})\,X^{k}$ where
$\alpha_{j,k}\in\mu_{n}\cup\\{0\\}$. Since $Y^{m}=X$ one obtains the unique
finite decomposition

$z=\sum_{j,k}\iota(\alpha_{j,k})Y^{j+mk},\qquad\alpha_{j,k}\in\mu_{n}\cup\\{0\\}.$

∎

The following example is a straightforward generalization of the fact that $3$
is an ${{{\mathbb S}}[\pm 1]}={{\mathbb S}}[\mu_{2,+}]$-generator of the ring
${\mathbb Z}$ of integers.

######  Example 5.6.

Let $m\in{\mathbb N}$ be a positive integer, and $\epsilon=\pm 1$. Then
$X=(3\epsilon)^{1/m}$ is an ${{{\mathbb S}}[\pm 1]}$-generator of the subring
$R={\mathbb Z}[X]$ of the number field ${\mathbb Q}((3\epsilon)^{1/m})$.

Indeed, the polynomial $X^{m}-3\epsilon$ is irreducible, thus every element
$z\in R$ can be written uniquely as a sum

$z=\sum_{j=0}^{m-1}a_{j}X^{j},\qquad a_{j}\in{\mathbb Z}.$

In turns, every $a_{j}$ can be uniquely written as
$a_{j}=\sum_{j,k}\alpha_{j,k}\,(3\epsilon)^{k}$, where
$\alpha_{j,k}\in\\{-1,0,1\\}$. Since $3\epsilon=X^{m}$ one obtains the unique
decomposition

$z=\sum_{j,k}\alpha_{j,k}X^{j+mk},\qquad\alpha_{j,k}\in\\{-1,0,1\\}.$

An interesting case is for $m=2$ and $\epsilon=-1$ since then the ring
$R={\mathbb Z}[\sqrt{-3}]$ is an order of the ring of integers of the
imaginary quadratic field ${\mathbb Q}(\sqrt{-3})$.

Notice that in the Example 5.6 the addition is specified by an equality of the
following form

$1+1=P(X),\qquad
P(X)=\sum_{j}\alpha_{j}\,X^{j},\qquad\alpha_{j}\in\\{-1,0,1\\},$ (5.7)

with $P(X)=\epsilon\,X^{m}-1$. A simple algebraic presentation of the form
(5.7) holds when working over $\mu_{n,+}$ for $n=1,2$.

The following result states the uniqueness of a similar polynomial
presentation in the general case.

######  Proposition 5.8.

Let $R$ be a ring, $\iota:\mu_{n}\to R^{\times}$ be an injective group
homomorphism, $X\in R$ an ${{\mathbb S}}[\mu_{n,+}]$-generator of $R$. For a
polynomial decomposition $z=\sum_{j}\iota(\alpha_{j})\,X^{j}\in R$, let
$\deg(z)$ be the smallest integer $m$ such that $\alpha_{j}=0$ for all $j>m$.
Then, the following results hold

  1. (i)

Let $m\in{\mathbb N}$, and $J_{m}=\langle X^{m}\rangle\subset R$ be the ideal
generated by $X^{m}$. Any element $z\in R$ admits a unique decomposition as
$z=a+b$ where $\deg(a)<m$ and $b\in J_{m}$.

  2. (ii)

The quotient $R_{m}:=R/J_{m}$ is a finite ring whose elements are uniquely
written as $\sum_{j=0}^{m-1}\iota(\alpha_{j})\,X^{j}$, with
$\alpha_{j}\in\mu_{n,+}=\mu_{n}\cup\\{0\\}$.

  3. (iii)

The quotient $R_{1}:=R/J_{1}$ is a finite field with $n+1$ elements and
$\iota:\mu_{n,+}\to R$ is a multiplicative section of the quotient map $R\to
R_{1}$.

  4. (iv)

The canonical ring homomorphism $\pi:R\to\varprojlim R_{m}$ is injective.

  5. (v)

The pair $(R,X)$ is uniquely specified, up to isomorphism, by the map
$h:\mu_{n}\to{\mathcal{P}}(\mu_{n})$ which is uniquely defined by the equality
$\sigma(h(\xi))=\iota(\xi)+1$.

###### Proof.

$(i)$ Let $z=\sum_{j}\iota(\alpha_{j})\,X^{j}$. By writing $z$ as

$z=\sum_{j=0}^{m-1}\iota(\alpha_{j})\,X^{j}+\sum_{j=m}^{\deg(z)}\iota(\alpha_{j})\,X^{j}=a+X^{m}c$
(5.9)

one obtains the required decomposition with $b=X^{m}\,c$. The uniqueness of
such decomposition then follows from the uniqueness of the decomposition as in
Definition 5.1.
$(ii)$ Follows from $(i)$. In particular, one easily checks that $R_{m}$ has
cardinality $\\#(R_{m})=(n+1)^{m}$.
$(iii)$ By construction the map $\iota:\mu_{n,+}\to R$ is a multiplicative
section of the quotient map $R\to R_{1}$. It follows that the non-zero
elements of $R_{1}$ form the multiplicative group $\mu_{n}$ so that $R_{1}$ is
a field with $n+1$ elements.
$(iv)$ The components of $z=\sum_{j}\iota(\alpha_{j})\,X^{j}\in R$ are
uniquely determined by $\pi(x)$.
$(v)$ Let $(R^{\prime},X^{\prime})$ be a second pair corresponding to the same
map $h:\mu_{n}\to{\mathcal{P}}(\mu_{n})$. Let $\rho:R\to R^{\prime}$ be the
bijective map defined by

$\rho\Big{(}\sum_{j}\iota(\alpha_{j})\,X^{j}\Big{)}:=\sum_{j}\iota^{\prime}(\alpha_{j})\,X^{\prime
j},\qquad\alpha_{j}\in\mu_{n}\cup\\{0\\}.$

One has by construction

$\deg(a)<m~{}\Longrightarrow~{}\rho(a+X^{m}b)=\rho(a)+(X^{\prime})^{m}\rho(b),\qquad\forall
b.$ (5.10)

In particular one also has $\rho(J_{m})=J^{\prime}_{m}$ for all $m$. Thus
$\rho$ induces a bijection $\rho_{m}:R_{m}\to R^{\prime}_{m}$. By $(iii)$, to
show that $\rho$ is a ring homomorphism, it is enough to verify that each
$\rho_{m}$ is a ring homomorphism.
To show that $\rho_{m}$ is additive it is enough to show that one can compute
all the components of a sum

$\sum_{j=0}^{m-1}\alpha_{j}\,X^{j}+\sum_{j=0}^{m-1}\beta_{j}\,X^{j}=\sum_{j=0}^{m-1}\gamma_{j}\,X^{j}$
(5.11)

using only the map $h:\mu_{n}\to{\mathcal{P}}(\mu_{n})$. To do this one first
determines a map $F$ from $k$-tuples of elements of elements of $\mu_{n,+}$ to
pairs $(x,Z)$ where $x\in\mu_{n,+}$ and where $Z$ is a $(k-1)$-tuple of
elements of ${\mathcal{P}}(\mu_{n})$. The map $h$ determines uniquely a
symmetric map

$H:\mu_{n,+}\times\mu_{n,+}\to\mu_{n,+}\times{\mathcal{P}}(\mu_{n}),\ \
H(\xi,\eta)=(\xi+\eta,0)\ \ \text{if}\ \ \xi\ \eta=0$
$H(\xi,\eta)=(H_{0}(\xi,\eta),P(\xi,\eta)),\ \
H_{0}(\xi,\eta)+XP(\xi,\eta)=\eta\ h(\xi\ \eta^{-1})\ \text{if}\ \ \eta\neq 0$
(5.12)

To define $F$ one proceeds by induction on $k$. For $k=1$ one lets $F(x)=x$.
For $k=2$ one lets $F_{2}=H$. We denote the two components of
$F_{k}:\mu_{n,+}^{k-1}\times\mu_{n,+}\to\mu_{n,+}\times{\mathcal{P}}(\mu_{n})^{k-1}$
as $F_{k}^{(1)}$ and $F_{k}^{(2)}$. To pass from $k-1$ to $k$ one lets

$F^{(1)}_{k}(\alpha,\eta):=(H_{0}(F^{(1)}_{k-1}(\alpha),\eta),\ \
F_{k}^{(2)}(\alpha,\eta):=(F^{(2)}_{k-1}(\alpha),P(F^{(1)}_{k-1}(\alpha),\eta))$

where in the last expression we append the polynomial
$P(F^{(1)}_{k-1}(\alpha),\eta)$ to the list $F^{(2)}_{k-1}(\alpha)$, thus
obtaining a list of $k-1$ polynomials.
To compute the components $\gamma_{j}$ of the sum (5.11), we build by
induction on $k$, two lists. The first $R(k)$ is the list of the coefficients
already computed and it is the single list given by
$(\gamma_{0},\gamma_{1},\ldots,\gamma_{k-1})$. The second $C(k)$, (called the
carry), is a list of polynomials with coefficients in $\mu_{n,+}$ and it is
encoded as the list of their coefficients. Each such list $\ell$ of
coefficients has $m-k$ terms, all in $\mu_{n,+}$. We denote by
$f(\ell)\in\mu_{n,+}$ the first term of the list $\ell$ and by $t(\ell)$ the
list obtained by dropping the first element of the list $\ell$, it has $m-k-1$
terms. The step to obtain $R(k+1),C(k+1)$ from $\alpha,\beta,R(k),C(k)$ is

$R(k+1):=F^{(1)}(\alpha_{k},\beta_{k},(f(\ell))_{\ell\in C(k)})$

and

$C(k+1):=(t(\ell))_{\ell\in
C(k)},F^{(2)}(\alpha_{k},\beta_{k},(f(\ell))_{\ell\in C(k)})$

where one replaces each element of
$F^{(2)}(\alpha_{k},\beta_{k},(f(\ell))_{\ell\in C(k)})$ by the list of its
first $m-k$ coefficients.
More concretely one first obtains
$\gamma_{0}=F^{(1)}_{2}(\alpha_{0},\beta_{0})$ while the carry over delivers
the polynomial $P(\alpha_{0},\beta_{0})=F^{(2)}_{2}(\alpha_{0},\beta_{0})$.
Thus $R(1)=(\gamma_{0})$, $C(1)$ has one element which is the list of the
first $m-1$ coefficients of $P(\alpha_{0},\beta_{0})$. One then trims the
elements $\alpha,\beta$ to and considers the sum

$\sum_{j=1}^{m-1}\alpha_{j}\,X^{j}+\sum_{j=1}^{m-1}\beta_{j}\,X^{j}+XP(\alpha_{0},\beta_{0})$
(5.13)

All terms in (5.13) are divisible by $X$ and one can use $F_{3}$ to compute
the sum of the three terms $\alpha_{1},\beta_{1},p_{0}$ where $p_{0}$ is the
constant term of $P(\alpha_{0},\beta_{0})$. This operation delivers the next
term

$\gamma_{1}=F^{(1)}_{3}(\alpha_{1},\beta_{1},p_{0})$

of (5.11), and adjoins the two polynomials of the list
$F^{(2)}_{3}(\alpha_{1},\beta_{1},p_{0})$ to the list of carry over consisting
of the single polynomial $P(\alpha_{0},\beta_{0})$ with its first term $p_{0}$
deleted. The carry over list consists now of three terms
$\ell_{1},\ell_{2},\ell_{3}$. One then uses $F_{5}$ to compute the sum of the
$5$ terms : $\alpha_{2},\beta_{2}$ and the three terms
$f(\ell_{1}),f(\ell_{2}),f(\ell_{3})$ from the carry over. This adds $4$ terms
to the list of carry over which has now $7$ terms, where the three previous
ones have been trimmed by deleting their lowest term. After $k$ such steps the
carry over list has $2^{k}-1$ elements and one proceeds as follows. One uses
$F_{2^{k}+1}$ to compute the sum of the $2^{k}+1$ terms given by
$\alpha_{k},\beta_{k}$ together with the terms $f(\ell)$ of the carry over
list. This operation delivers $\gamma_{k}$ and adjoins $2^{k}$ terms to the
carry over list which now consists of $2^{k+1}-1$ terms. This process
terminates when $k=m$ and $R(m)$ delivers universal formulas for the terms
$\gamma_{j}$, $0\leq j\leq m-1$ using only $\alpha,\beta$ and the map $h$.
The fact that the coefficients $\gamma_{j}$ can be computed using only
$\alpha,\beta$ and the map $h$ proves that $\rho$ is additive since one can
use the same formula to compute $\alpha+\beta$ in $R_{m}$ and
$\rho_{m}(\alpha)+\rho_{m}(\beta)$ in $R^{\prime}_{m}$. The multiplicativity
of $\rho$ follows by bilinearity from $\rho(\alpha X^{n}\times\beta
X^{m})=\rho(\alpha X^{n})\rho(\beta X^{m})$. This shows that $\rho:R\to
R^{\prime}$ is a ring isomorphism and by construction one has
$\rho(X)=X^{\prime}$.∎

######  Definition 5.14.

The map

$h:\mu_{n}\to{\mathcal{P}}(\mu_{n}),\qquad\sigma(h(\xi))=\iota(\xi)+1$ (5.15)

which characterizes the pair $(R,X)$ (by Proposition 5.8) is called the hold
of the pair $(R,X)$.

######  Corollary 5.16.

Let $n$ be such that there exists a polynomial ring in one generator over
${{\mathbb S}}[\mu_{n,+}]$, then $n+1$ is a prime power.

###### Proof.

This follows from Proposition 5.8 $(iii)$. ∎

######  Remark 5.17.

The proof of Proposition 5.8 $(v)$ is stated so that one can, by following it,
write a computer program which can be used to test the additive structure of
the ring $R_{m}$. This will be relevant in section 6 to determine in the
various examples the rings $R_{m}$.
The map $h:\mu_{n}\to{\mathcal{P}}(\mu_{n})$ of (5.15) determines the addition
$H:\mu_{n,+}\times\mu_{n,+}\to\mu_{n,+}\times{\mathcal{P}}(\mu_{n})$, (5.12),
of pairs of elements of $\mu_{n,+}$ using the compatibility with
multiplication by elements of $\mu_{n}$.
Proposition 5.8 shows that a pair $(R,X)$, where $X$ is an ${{{\mathbb S}}[\pm
1]}$-generator of $R$, i.e. $n=2$, is uniquely characterized by the polynomial
$P(X)$ as in (5.7). The polynomial $P(X)=-1$ produces the pair $({\mathbb
F}_{3}[X],X)$, while $P(X)=X-1$ determines the pair $({\mathbb Z},3)$.
When $n=2$, the constant term of the polynomial $P(X)$ in (5.7) is necessarily
equal to $-1$. Indeed, had the constant term be $0$ or $1$, one would
contradict the uniqueness of the decomposition of Definition 5.1 by the
equality $1=P(X)-1$. This also shows that $R_{1}={\mathbb F}_{3}$.

######  Remark 5.18.

It is not true that a random choice of a polynomial with coefficients in
${{{\mathbb S}}[\pm 1]}$ and constant coefficient $-1$ corresponds to a pair.
A simple case is with $P(X)=-1+X+X^{2}$. Indeed, in the following lines we
show that $5$ is not represented by any polynomial. With this rule, one has
$1+1+1+1=1+X+X^{2}$. Adding $1$ to both sides gives

$\displaystyle 1+1+1+1+1=-1+X+X^{2}+X+X^{2}=-1+X(-1+X+X^{2})+X^{2}(-1+X+$
$\displaystyle+X^{2})=-1-X+X^{3}+X^{3}+X^{4}=-1-X+X^{3}(-1+X+X^{2})+X^{4}=$
$\displaystyle=-1-X-X^{3}+X^{4}+X^{4}+X^{5}.$

Then, when working in $R_{n}$ (i.e. modulo $X^{n}$) the number $5$ is
represented by

$5=-1-X-X^{3}-X^{4}-X^{5}-\cdots-X^{n-1}\in R_{n}$

and this expression is of degree $n-1$ for any $n$ and thus does not
correspond to a finite sum of powers of $X$.

##  6 Examples

In this section we give examples of polynomial rings $(R,X)$ in one generator
$X$ over ${{\mathbb S}}[\mu_{n,+}]$ where $R$ is of characteristic zero. The
ring $R$ is embedded as a subring of ${\mathbb C}$ by solving for
$X\in{\mathbb C}$ the equations $\sigma(h(\xi))=\iota(\xi)+1,\,\xi\in\mu_{n}$,
using the canonical embedding $\mu_{n,+}\subset{\mathbb C}$. The projective
limit $\varprojlim R_{n}$ is, in these examples, a finite extension of the
ring of $p$-adic integers ${\mathbb Z}_{p}$. While one can use the axiom of
choice to show the existence of an embedding of the $p$-adic field ${\mathbb
Q}_{p}$ in the field of complex numbers, such embeddings have the status of a
chimera. Indeed, the continuity of measurable characters of compact groups
applied to the additive group ${\mathbb Z}_{p}$ shows that an embedding of the
$p$-adic field ${\mathbb Q}_{p}$ in the field of complex numbers is
automatically non-measurable. On the other hand, next examples will show that
polynomial rings $(R,X)$ in one generator $X$ over ${{\mathbb S}}[\mu_{n,+}]$
provide instances of explicit interactions of $p$-adic fields (and their
finite extensions) with the complex numbers. These interactions are given by
pairs of embeddings with dense ranges

$\begin{array}[c]{ccccccccc}{\mathbb
F}_{q}&\stackrel{{\scriptstyle\pi}}{{\twoheadleftarrow}}&W({\mathbb
F}_{q})&\hookrightarrow&W({\mathbb
F}_{q})[\eta]&\hookleftarrow&R[X^{-1}]&\hookrightarrow&{\mathbb C}\end{array}$

of the ring of Laurent polynomials $R[X^{-1}]$. The left embedding in the
above diagram is in a finite algebraic extension $W({\mathbb F}_{q})[\eta]$ of
the Witt ring $W({\mathbb F}_{q})$. The field of fractions of the ring
$W({\mathbb F}_{q})[\eta]$ is a finite extension of the $p$-adic field. Most
of these examples come from known number systems and have their origin in the
search of optimal manners of encoding numbers [20]. In each case, the quotient
$R_{1}=R/(XR)$ is the finite field ${\mathbb F}_{q}$, $q=n+1$, and the
multiplicative semi-group isomorphism $j:{\mathbb
F}_{q}\sim\mu_{n,+}\subset{\mathbb C}$ serves as a guide, using the addition
in the finite field ${\mathbb F}_{q}$, for the terms of degree $0$ in the
construction of the map $h$. Note that the choice of $j$ for $\bar{\mathbb
F}_{q}$ plays a key role in the construction by Quillen [27] of the relation
between the algebraic $K$-theory of ${\mathbb F}_{q}$ and the Adams
operations.

###  6.1  Polynomial rings in one generator over ${{\mathbb S}}={{\mathbb
S}}[\mu_{1,+}]$

When working over ${{\mathbb S}}={{\mathbb S}}[\mu_{1,+}]$ there is no
cancellation since there is no minus sign available. Thus starting from two
non-zero elements $x,y$ the equality $x+y=0$ can only be verified in the
projective limit $\varprojlim R_{m}$. We compute this projective limit in the
next examples.

####  6.1.1 The polynomial ring $({\mathbb Z},-2)$

The ring ${\mathbb Z}$ admits the generator $X=-2$ over ${{\mathbb S}}$. The
hold is given by $1+1=P(X)=X+X^{2}$. The values of the polynomials of degree
$n$, at $X=-2$ are reported for the first values of $n$ in the following table

$\begin{array}[]{|c |c|}\hline\cr n&\\{p(-2):\deg p=n\\}\\\ \hline\cr
0&[0,1]\cap{\mathbb Z}\\\ 1&[-2,-1]\cap{\mathbb Z}\\\ 2&[2,5]\cap{\mathbb
Z}\\\ 3&[-10,-3]\cap{\mathbb Z}\\\ 4&[6,21]\cap{\mathbb Z}\\\
5&[-42,-11]\cap{\mathbb Z}\\\ 6&[22,85]\cap{\mathbb Z}\\\
\hline\cr\end{array}$

Let us look, for example, at the computation of $1+1+X$. One gets

$1+1+X=X+X^{2}+X=X(1+1+X)$

and iterating this step one gets that $1+1+X\in J_{m}=\langle X^{m}\rangle R$,
$\forall m$. This shows that $1+1+X=0$ in $\varprojlim R_{m}$. Next we relate
the degree of the polynomial $p(X)$ with the absolute value of the integer
$p(-2)$. Let

$j(n):=\frac{1}{3}(-2)^{n}-\frac{1}{2}(-1)^{n}+\frac{1}{6}\qquad n\in{\mathbb
N}.$ (6.1)

The degree $n$ of a polynomial $p(X)$ with coefficients in $\\{0,1\\}$
specifies the sign of the integer $p(-2)$ as $(-1)^{n}$ and provides lower and
upper bounds on the modulus $|p(-2)|$ as follows

$|j(n-1)|<|p(-2)|\leq|j(n+1)|.$

Given an integer $m\in{\mathbb Z}$, the first inequality provides the
following bound, on the degree of the polynomial $p$ such that $p(-2)=m$

$\deg(p)\leq\log_{2}(3|m|+2)+1.$

The projective limit $\varprojlim R_{m}$ is here the ring ${\mathbb Z}_{2}$ of
$2$-adic integers, and the elements of ${\mathbb Z}$ inside ${\mathbb Z}_{2}$
are characterized by the fact that their sequence of digits is eventually
constant.

Next, we turn to quadratic fields for which the study of number systems in
[18, 19] provides an exhaustive list of examples. One easily deduces from
op.cit. the following

######  Proposition 6.2.

The quadratic fields $K$ whose ring of integers admit an ${{\mathbb
S}}$-generator are

  * •

${\mathbb Q}(\sqrt{-1})$ with generator $X=-1+\sqrt{-1}$ of the ring ${\mathbb
Z}[\sqrt{-1}]$ of integers of $K$.

  * •

${\mathbb Q}(\sqrt{-2})$ with generator $X=\sqrt{-2}$ of the ring ${\mathbb
Z}[\sqrt{-2}]$ of integers of $K$.

  * •

${\mathbb Q}(\sqrt{-7})$ with generator
$X=\frac{1}{2}\left(1+\sqrt{-7}\right)$ of the ring of integers of $K$.

###### Proof.

The norm $N(\alpha)$ of an ${{\mathbb S}}$-generator is equal to $2$, thus the
set $N_{0}(\alpha):=\\{0,\ldots,N(\alpha)-1\\}$ defining a canonical number
system in the sense of [18, 19] is $\\{0,1\\}$ and the result follows from
Theorem 1 of [19] in the complex case and Satz 1 of [18] in the real case.∎

####  6.1.2 The polynomial ring $({\mathbb Z}[i],-1+i)$

Here, we consider the ring $R={\mathbb Z}[i]$ of Gaussian integers (sometimes
called binarions; see [22]) with $X=-1+i$ as ${{\mathbb
S}}[\mu_{1,+}]={{\mathbb S}}$-generator. Indeed, every Gaussian integers can
be written uniquely as a finite sum of powers of $X$ ([16, 28] and Figure 2).
One has the equality $1+1=P(X)=X^{2}+X^{3}$, which allows one to compute the
sum of any pair of polynomials with coefficients in $\\{0,1\\}$.

Figure 2: Gaussian integers as ${{\mathbb S}}$-polynomials in degree $\leq 12$

######  Proposition 6.3.

Let $R={\mathbb Z}[i]$, $X=-1+i$.
$(i)$ The ideal of $R={\mathbb Z}[i]$ generated by $X^{2}$ is the same as the
ideal generated by $2$.
$(ii)$ The ring $R_{m}$ for $m=2k$ is ${\mathbb Z}/(2^{k}{\mathbb Z})[X]$
where $X^{2}=-2-2X$.
$(iii)$ The ring $R_{m}$ for $m=2k+1$ is ${\mathbb Z}/(2^{k+1}{\mathbb
Z})\oplus{\mathbb Z}/(2^{k}{\mathbb Z})\,X$ where $X^{2}=-2-2X$.
$(iv)$ The projective limit $\varprojlim R_{m}$ is the ring ${\mathbb
Z}_{2}[i]\sim{\mathbb Z}_{2}[X]$ where $X^{2}=-2-2X$.

###### Proof.

$(i)$ The element $U=(1+X)\in R$ is a unit since $U^{4}=1$ and one has

$X^{2}=-2-2X\in 2R,\ \ 2=-(1+X)^{-1}X^{2}\in X^{2}R$

$(ii)$ By $(i)$ the ideal $X^{2k}R$ is equal to $2^{k}R$. One has $R={\mathbb
Z}[i]$ and $R/(2^{k}R)={\mathbb Z}/(2^{k}{\mathbb Z})[i]={\mathbb
Z}/(2^{k}{\mathbb Z})[X]$ with $X^{2}=-2-2X$, thus one gets $(ii)$.
$(iii)$ Let $m=2k+1$. Any element of $R$ is of the form $z=a+bX$ where
$a,b\in{\mathbb Z}$. In $R$ one has $2^{k+1}\in X^{2k+2}R\subset J_{m}$ and
$2^{k}X\in X^{2k+1}R=J_{m}$. Thus the homomorphism ${\mathbb Z}[X]\to R_{m}$
induces a surjective homomorphism from ${\mathbb Z}/(2^{k+1}{\mathbb
Z})\oplus{\mathbb Z}/(2^{k}{\mathbb Z})\,X$ to $R_{m}$. It is bijective since
the cardinalities are equal.
$(iv)$ The extension ${\mathbb Q}_{2}[i]$ is totally ramified of index $e=2$
(see [28], 4.2). The polynomial $X^{2}+2X+2$ is an Eisenstein polynomial which
defines ${\mathbb Q}_{2}[i]$ as its splitting field. The valuation of $X$ is
one half of the valuation of $2$.∎

####  6.1.3 The polynomial ring $\left({\mathbb
Z}[\sqrt{-2}],\sqrt{-2}\right)$

The element $X:=\sqrt{-2}$ is an ${{\mathbb S}}[\mu_{1,+}]={{\mathbb
S}}$-generator of the ring of integers ${\mathbb Z}[\sqrt{-2}]$ of the
imaginary quadratic field ${\mathbb Q}(\sqrt{-2})$. This follows directly from
§6.1.1 and Proposition 5.5. The hold is given by the polynomial
$P(X)=X^{4}+X^{2}$. A straightforward analogue of Proposition 6.3 holds.

####  6.1.4 The polynomial ring $\left({\mathcal{O}}({\mathbb
Q}(\sqrt{-7})),\frac{1}{2}(1+\sqrt{-7})\right)$

The element $X:=\frac{1}{2}\left(1+\sqrt{-7}\right)$ is an ${{\mathbb
S}}[\mu_{1,+}]={{\mathbb S}}$-generator of the ring ${\mathcal{O}}({\mathbb
Q}(\sqrt{-7}))$ of integers of the imaginary quadratic field ${\mathbb
Q}(\sqrt{-7})$. The hold is given by the polynomial $P(X)=X^{3}+X$. Let $F$ be
the fundamental domain of ${\mathcal{O}}({\mathbb Q}(\sqrt{-7}))$ given by the
parallelogram with vertices $0,1,X,X+1$. Figure 3 shows the neighborhood of
$0\in{\mathbb C}$ obtained as the union of the translations $F+p(X)$ by
polynomials $p(X)$ of degree $\leq 11$.

Figure 3: Polynomials of degree $\leq 11$ for
$X=\frac{1}{2}\left(1+\sqrt{-7}\right)$

######  Proposition 6.4.

Let $R={\mathcal{O}}({\mathbb Q}(\sqrt{-7}))$,
$X=\frac{1}{2}\left(1+\sqrt{-7}\right)$.
$(i)$ The ring $R_{m}$ is ${\mathbb Z}/(2^{m}{\mathbb Z})$.
$(ii)$ The projective limit $\varprojlim R_{m}$ is the ring ${\mathbb Z}_{2}$.
$(iii)$ The element $X\in\varprojlim R_{m}={\mathbb Z}_{2}$ is the only
solution divisible by $2$ in the ring ${\mathbb Z}_{2}$ for the equation
$2+X+X^{2}=0$.

###### Proof.

The hold is given by $P(X)=X^{3}+X$ and one has
$P(X)-2=(X-1)\left(X^{2}+X+2\right)$. By Hensel’s Lemma, the equation
$2+X+X^{2}=0$ admits a unique solution $\alpha$ in ${\mathbb Z}_{2}$ of the
form $\alpha=1+2\epsilon$ and a unique solution of the form
$\beta=2(1+2\epsilon^{\prime})$. In fact one has $\alpha\beta=2$ and
$\alpha+\beta=-1$. The homomorphism $\rho:{\mathbb
Z}[\frac{1}{2}\left(1+\sqrt{-7}\right)]\to{\mathbb Z}_{2}$ given by
$\rho\left(\frac{1}{2}\left(1+\sqrt{-7}\right)\right)=\beta$ is well defined
since $\beta$ is a solution of the equation $2+X+X^{2}=0$. Moreover $\beta$ is
the product of $2$ by a unit of ${\mathbb Z}_{2}$ (but this fails in
$R={\mathcal{O}}({\mathbb Q}(\sqrt{-7}))$). The projection $X_{m}$ of $\beta$
in ${\mathbb Z}_{2}/(2^{m}{\mathbb Z}_{2})={\mathbb Z}/(2^{m}{\mathbb Z})$
fulfills $P(X_{m})=2$ and $X_{m}$ is the product of $2$ by a unit. Thus the
ideals generated by powers of $X_{m}$ are the same as those generated by
powers of $2$. This proves the three assertions $(i)$, $(ii)$, $(iii)$. ∎

###  6.2 Polynomial rings in one generator over ${{{\mathbb S}}[\pm 1]}$

####  6.2.1 The polynomial ring $({\mathbb Z},3)$

The case of the ${{{\mathbb S}}[\pm 1]}$-generator $3\in{\mathbb Z}$ is
particularly relevant because, as shown in [13], the addition coincides with
that of the Witt vectors in $W({\mathbb F}_{3})={\mathbb Z}_{3}$.

######  Proposition 6.5.

Let $R={\mathbb Z}$, $X=3$ is an ${{{\mathbb S}}[\pm 1]}$-generator of $R$.
The hold is $P(X)=-1+X$.
$(i)$ The ring $R_{m}$ is ${\mathbb Z}/(3^{m}{\mathbb Z})$.
$(ii)$ The projective limit $\varprojlim R_{m}$ is the ring $W({\mathbb
F}_{3})={\mathbb Z}_{3}$.
$(iii)$ The set of Witt vectors with only finitely many non-zero components
forms a subring of $W({\mathbb F}_{3})$ isomorphic to ${\mathbb Z}$.

In order to organize the next examples we give the list of imaginary quadratic
field extensions of ${\mathbb Q}$ generated by rings of ${{{\mathbb S}}[\pm
1]}$-polynomials in one variable.

######  Proposition 6.6.

The imaginary quadratic fields $K$ generated by rings of ${{{\mathbb S}}[\pm
1]}$-polynomials in one variable are

  * •

${\mathbb Q}(\sqrt{-2})$ with generator $X=1+\sqrt{-2}$ of the ring ${\mathbb
Z}[\sqrt{-2}]$ of integers of $K$.

  * •

${\mathbb Q}(\sqrt{-3})$ with generator $X=\sqrt{-3}$ of ${\mathbb
Z}[\sqrt{-3}]$ (not a UFD).

  * •

${\mathbb Q}(\sqrt{-11})$ with generator $X=\frac{1}{2}(1+\sqrt{-11})$ of the
ring of integers of $K$.

###### Proof.

Let $P(X)=-1+\sum_{j=1}^{n-1}a(j)X^{j}+\epsilon X^{n}$, $\epsilon\in\\{\pm
1\\}$, $a(j)\in\\{-1,0,1\\}$, be the carry leading to an imaginary quadratic
extension. The roots of the polynomial $P(X)-2$ are algebraic integers, and we
assume that one of them, say $\alpha$, is quadratic imaginary. Let
$q(x)=x^{2}-bx+c$ be its minimal polynomial. It has integral coefficients so
$b,c\in{\mathbb Z}$, and by definition, it divides $P(X)-2$. The constant
coefficient $c$ must be equal to $3$. Indeed it divides the constant
coefficient $-3$ of $P(X)-2$ and since $b^{2}-4c<0$ it is positive. It cannot
be equal to $1$ since in that case one would get $b\in\\{-1,0,1\\}$, and
$\alpha\in\\{i,j,-j\\}$ which contradicts the injectivity of the map $\sigma$.
For $c=3$ the possible values of $b$ are $b=0$ which gives the solution
$\alpha=\sqrt{-3}$, $b=\pm 1$ which gives the solutions
$\alpha=\frac{1}{2}\left(\pm 1\pm i\sqrt{11}\right)$, $b=\pm 2$ which gives
the solutions $\alpha=\pm 1\pm i\sqrt{2}$, and finally $b=\pm 3$. We shall now
show that this last choice which gives $\alpha=\frac{1}{2}\left(\pm 3\pm
i\sqrt{3}\right)$ does not give a solution. To prove this it is enough to show
that the polynomial $3+3X+X^{2}$ cannot divide a polynomial $P(X)-2$ with $P$
of the above form. We thus assume an equality of the form

$(3+3X+X^{2})\left(\sum_{j=0}^{n-2}b(j)X^{j}\right)=-3+\sum_{j=1}^{n-1}a(j)X^{j}+\epsilon
X^{n},\ \epsilon\in\\{\pm 1\\},\ a(j)\in\\{-1,0,1\\}$

Since the coefficients of $P-2$ are integers and the leading coefficient of
$3+3X+X^{2}$ is $1$ the coefficients $b(j)$ are integers. We get $b(0)=-1$,
$3b(1)-3=a(1)$, but $a(1)\in\\{-1,0,1\\}$ and thus working modulo $3$ one gets
$a(1)=0$ and hence $b(1)=1$. Considering the coefficient of $X^{2}$ we get
$3b(1)+3b(2)-1=a(2)$ which gives $a(2)=-1$ and $b(2)=-b(1)=-1$. We can now
work by induction to show that $b(j)=(-1)^{j+1}$. Indeed the coefficient of
$X^{j}$ is $b(j-2)+3b(j-1)+3b(j)=a(j)$ and if we know that $b(j-2)=(-1)^{j-1}$
and $b(j-1)=(-1)^{j}$ we get $a(j)=b(j-2)$ and $3b(j-1)+3b(j)=0$ so that
$b(j)=(-1)^{j+1}$. This works for $j\leq n-2$. The coefficient of $X^{n-1}$ is
$b(n-3)+3b(n-2)=a(n-1)$ and this gives a contradiction since one gets
$a(n-1)=b(n-3)$ (working modulo 3) which contradicts the fact that $b(n-2)\neq
0$. ∎

####  6.2.2 The polynomial ring $({\mathcal{O}}({\mathbb
Q}[\sqrt{-11}]),\frac{1}{2}\left(1+\sqrt{-11}\right))$

This section is dedicated to a detailed proof that
$X:=\frac{1}{2}\left(1+\sqrt{-11}\right)$ is an ${{{\mathbb S}}[\pm
1]}$-generator of the ring of integers of the number field ${\mathbb
Q}(\sqrt{-11})$. The reason for providing the details of the proof is because
we want to emphasize that in such a case, and unlike working over ${{\mathbb
S}}$, one can explicitly control the cancellations in the computations.

######  Proposition 6.7.

Let ${\mathcal{O}}$ be the ring of integers of the number field ${\mathbb
Q}(\sqrt{-11})$.
$(i)$ $X:=\frac{1}{2}\left(1+\sqrt{-11}\right)$ is an ${{{\mathbb S}}[\pm
1]}$-generator of ${\mathcal{O}}$. The hold of $({\mathcal{O}},X)$ is
$P(X)=-1+X-X^{2}$.
$(ii)$ The projective limit $\varprojlim R_{m}$ is the ring $W({\mathbb
F}_{3})={\mathbb Z}_{3}$.

Figure 4: Fundamental domain of the lattice ${\mathcal{O}}$

The proof requires a preliminary lemma. We first recall some classical results
concerning the ring of integers ${\mathcal{O}}$ of the imaginary quadratic
field $K={\mathbb Q}(\sqrt{-11})$. The discriminant of $K$ is $d=-11$. Thus
since $-11\sim 1$ modulo $4$, the lattice ${\mathcal{O}}$ is ${\mathbb
Z}+{\mathbb Z}X$ where $X:=\frac{1}{2}\left(1+\sqrt{-11}\right)$. By
construction one has

$1+1=P(X),\qquad P(X)=-1+X-X^{2}.$ (6.8)

One wants to show that every element $z\in{\mathcal{O}}$ can be written
uniquely as a polynomial $z=\sum_{j}\alpha_{j}\,X^{j}$, with
$\alpha_{j}\in\\{-1,0,1\\}$. Figure 4 shows the translates of the fundamental
domain of the lattice, while the next figures provide a sketch of a few steps
of the process of representing elements of ${\mathcal{O}}$ in terms of
polynomials of degree $\leq n$, showing those described by polynomials of
degree $=n$ with a new color.

(a) First step, polynomials of degree $0$

(b) Second step, polynomials of degree $\leq 1$

Figure 5: The first two steps

(a) Third step, polynomials of degree $\leq 2$

(b) Fourth step, polynomials of degree $\leq 3$

Figure 6: The third and fourth steps

(a) Fifth step, polynomials of degree $\leq 4$

(b) Eigth’s step, polynomials of degree $\leq 7$

Figure 7: The fifth and eighth steps

By comparing Figures 5(a), 5(b), 6(a), 6(b), 7(a), 7(b), one notices that the
translation $z\mapsto z+1$ does not increase the degree of the polynomial by
more than $2$ units. Next lemma provides a formal proof of this fact.

######  Lemma 6.9.

Let $z=\sum_{j=0}^{n}\alpha_{j}X^{j}\in{\mathcal{O}}$,
$\alpha_{j}\in\\{-1,0,1\\}$. Then there exist coefficients
$\beta_{j}\in\\{-1,0,1\\}$, with $0\leq j\leq n+2$, such that
$z+1=\sum_{j=0}^{n+2}\beta_{j}X^{j}$.

###### Proof.

We proceed by induction on the integer $n$. For $n=0$, the result follows from
(6.8). Let us assume that the result is proved up to $n-1$, then there exists
coefficients $\gamma_{j}\in\\{-1,0,1\\}$ such that

$z=\left(\sum_{j=0}^{n-1}\alpha_{j}X^{j}\right)+\alpha_{n}X^{n}~{}\Longrightarrow~{}z+1=\left(\sum_{j=0}^{n+1}\gamma_{j}X^{j}\right)+\alpha_{n}X^{n}.$

Let us consider a sum such as
$\gamma_{n}X^{n}+\gamma_{n+1}X^{n+1}+\alpha_{n}X^{n}$ and express it without
going beyond $X^{n+2}$. If $\gamma_{n+1}=0$ this follows again from (6.8). We
can thus assume that $\gamma_{n+1}=\pm 1$ and also that both $\gamma_{n}$ and
$\alpha_{n}$ are non-zero and equal since otherwise the sum
$\gamma_{n}X^{n}+\alpha_{n}X^{n}$ would have degree at most $n$. The only case
to exclude then is when $\gamma_{n}$, $\alpha_{n}$, and $\gamma_{n+1}$ are all
equal (and non-zero), since only in that case would one get a term in
$X^{n+3}$ from the sum

$\displaystyle X^{n}+X^{n}+X^{n+1}=X^{n}(1+1+X)=X^{n}(-1+X+X-X^{2})=$
$\displaystyle=X^{n}(-1-X+X^{2}-X^{2}-X^{3})=-X^{n}-X^{n+1}-X^{n+3}.$

To exclude this case, one adds to the induction hypothesis the condition that
if the last term $\beta_{n+2}$ of the polynomial of degree $n+2$ representing
$z+1$ is non-zero, then the term $\beta_{n+1}$ is zero or of the opposite
sign. This condition is fulfilled for $n=0$, and if we assume it for $n-1$, it
holds also for $n$. Indeed, the only cases when $\beta_{n+2}\neq 0$ arise when
either $\gamma_{n+1}=0$, in which case $\beta_{n+1}$ and $\beta_{n+2}$ have
opposite signs, or $\gamma_{n+1}=\epsilon=\pm 1$ in which case
$\gamma_{n}=\alpha_{n}=-\epsilon$, which gives

$\gamma_{n}X^{n}+\gamma_{n+1}X^{n+1}+\alpha_{n}X^{n}=-\epsilon
X^{n}(1+1-X)=\epsilon X^{n}+\epsilon X^{n+2},$

implying that $\beta_{n+1}=0$ in this case. Thus the induction hypothesis
still holds for $n$, and this concludes the proof. ∎

###### Proof.

(of Proposition 6.7) Lemma 6.9 holds for the abstract law of addition defined
using (6.8) on the projective limit of the $R_{n}$. The proof shows that
elements of this limit, which have only a finite number of non-zero
coordinates, are stable under the addition of $1$. Using (5.10), it follows
that they are also stable under the addition of any monomial and hence that
they form an additive group $A$. Thus, it remains to show that the map
$\rho:A\to{\mathbb C}$ defined by

$\rho\Big{(}\sum_{j}\alpha_{j}X^{j}\Big{)}:=\sum_{j}\alpha_{j}z^{j},\qquad
z=\frac{1}{2}\left(1+\sqrt{-11}\right)$

is injective. Let $\sum_{j}\alpha_{j}X^{j}\in\ker\rho$, then
$\sum_{j}\alpha_{j}z^{j}=0$ and thus $z$ fulfills an equation $E(z)=0$ with
integral coefficients whose leading coefficient is $1$ and the constant term
is $\pm 1$. The polynomial $E$ is thus a multiple of the minimal polynomial
$z^{2}-z+3$ of the field extension. The quotient polynomial has integral
coefficients; thus, one gets a contradiction using the product of constant
terms.∎

####  6.2.3 The polynomial ring $\left({\mathbb
Z}[\sqrt{-3}],\sqrt{-3}\right)$

The element $X:=\sqrt{-3}$ is an ${{\mathbb S}}[\mu_{2,+}]={{{\mathbb S}}[\pm
1]}$-generator of the ring ${\mathbb Z}[\sqrt{-3}]$ and the latter is a
maximal order in the ring of integers of the imaginary quadratic field
${\mathbb Q}(\sqrt{-3})$. This follows directly from §6.1.1 and Proposition
5.5. The hold is given by the polynomial $P(X)=-1-X^{2}$. A straightforward
analogue of Proposition 6.3 holds.

####  6.2.4 The polynomial ring $\left({\mathcal{O}}({\mathbb
Q}(\sqrt{-2})),1+\sqrt{-2}\right)$

One obtains similarly that $P(X)=-1-X+X^{2}-X^{3}$ is the hold associated to
the ${{{\mathbb S}}[\pm 1]}$ generator $1+\sqrt{-2}$ of the ring of integers
of the imaginary quadratic field ${\mathbb Q}(\sqrt{-2})$

Figure 8: Polynomials of degree $\leq 9$ for $X=1+i\sqrt{2}$

######  Proposition 6.10.

Let ${\mathcal{O}}$ be the ring of integers of the number field ${\mathbb
Q}(\sqrt{-2})$.
$(i)$ $X:=1+\sqrt{-2}$ is an ${{{\mathbb S}}[\pm 1]}$-generator of
${\mathcal{O}}$. The hold of $({\mathcal{O}},X)$ is $P(X)=-1-X+X^{2}-X^{3}$.
$(ii)$ The projective limit $\varprojlim R_{m}$ is the ring $W({\mathbb
F}_{3})={\mathbb Z}_{3}$.

Figure 8 reproduces the pattern obtained by inputting polynomials of degree
$\leq 9$. In this case, the analog of Lemma 6.9 holds with the bound $n+3$
instead of $n+2$.

###  6.3 Polynomial rings in one generator over ${{\mathbb S}}[\mu_{3,+}]$

In the next example the field $R_{1}$ is the finite field ${\mathbb F}_{4}$.
One lets $\mu_{3,+}\subset{\mathbb C}$ be the solutions of $x(x^{3}-1)=0$,
$j=\exp(2\pi i/3)$ and ${\mathbb Z}(j)\subset{\mathbb Q}(j)$ be the ring of
integers of the quadratic imaginary field ${\mathbb Q}(j)$.

######  Proposition 6.11.

$(i)$ The number $-2\in{\mathbb Z}(j)$ is an ${{\mathbb
S}}[\mu_{3,+}]$-generator of the ring $R={\mathbb Z}(j)$.
$(ii)$ The hold is given by

$h(1)=X+X^{2},\ \ h(j)=j^{2}X+j^{2},\ \ h(j^{2})=jX+j$

$(iii)$ The field $R_{1}$ is the finite field ${\mathbb F}_{4}$.
$(iv)$ The projective limit $\varprojlim R_{m}$ is the Witt ring $W({\mathbb
F}_{4})$ and the ring $R_{m}$ is the quotient of $W({\mathbb F}_{4})$ by
$2^{m}\,W({\mathbb F}_{4})$.

###### Proof.

Let $J=2{\mathbb Z}(j)\subset{\mathbb Z}(j)$, then $J^{n}$ is the ideal
generated by $X^{n}$ where $X=-2$. Let $\sigma:{\mathcal{P}}(\mu_{3})\to
R={\mathbb Z}(j)$ be the map defined by (5.3). For each $n$ the composition
$\pi_{n}\circ\sigma$, from the subset
${\mathcal{P}}^{n-1}(\mu_{3})\subset{\mathcal{P}}(\mu_{3})$ formed of
polynomials of degree $<n$ to the quotient ring $R_{n}=R/J^{n}$, is surjective
and hence injective since the cardinalities of source and target are the same.
It follows that the map $\sigma:{\mathcal{P}}(\mu_{3})\to R={\mathbb Z}(j)$ is
injective. To show that it is surjective one uses the general method involving
the limit of the subsets

$Z_{n}:=(-2)^{-n}\left(\sigma({\mathcal{P}}^{n}(\mu_{3})+F)\right)\subset{\mathbb
C}$

where $F$ is a fundamental domain for ${\mathbb Z}(j)$. One observes that
passing from $n$ to $n+1$ only alters $Z_{n}$ on its boundary and that $Z_{n}$
contains an open disk centered at $0$.

Figure 9: Polynomials of degree $\leq 7$ for $X=-2$

###  6.4 Polynomial rings in one generator over ${{\mathbb S}}[\mu_{4,+}]$

######  Proposition 6.12.

$(i)$ The number $X=1+2i$ is an ${{\mathbb S}}[\mu_{4,+}]$-generator of the
ring $R={\mathbb Z}(i)$.
$(ii)$ The hold is given by $h(0)=1$ and

$h(1)=i-i\,X,\ \ h(i)=-i+X,\ \ h(-i)=-1-i\,X$

$(iii)$ The field $R_{1}$ is the finite field ${\mathbb F}_{5}$.
$(iv)$ The projective limit $\varprojlim R_{m}$ is the Witt ring $W({\mathbb
F}_{5})={\mathbb Z}_{5}$ and the ring $R_{m}$ is the quotient of $W({\mathbb
F}_{5})$ by $5^{m}\,W({\mathbb F}_{5})$.

###### Proof.

In the $p$-adic field ${\mathbb Z}_{5}$ there exists a unique square root of
$-1$ equal to $2$ modulo $5$ (see [28], §6.7). Let $\rho:{\mathbb
Z}(i)\to{\mathbb Z}_{5}$ be the unique morphism such that, modulo $5$, one has
$\rho(i)=2$. Then $\rho(X)=5u$ where $u$ is a unit in ${\mathbb Z}_{5}$. The
morphism $\rho$ restricted to $\mu_{4,+}=\\{0,1,i,-1,-i\\}$ gives a
multiplicative section of the quotient map $R\to R/XR$. One has ${\mathbb
Z}_{5}/\rho(X)^{m}{\mathbb Z}_{5}={\mathbb Z}/5^{m}{\mathbb Z}$ and the
morphism $\rho$ induces an isomorphism $R_{m}\simeq{\mathbb Z}_{5}={\mathbb
Z}/5^{m}{\mathbb Z}$. Statements $(iii)$ and $(iv)$ follow, as well as the
injectivity of the map $\sigma:{\mathcal{P}}(\mu_{4})\to R={\mathbb Z}(i)$.
One can prove the surjectivity of $\sigma$ as for Proposition 6.11 using
Figure 10. Statements $(i)$ and $(ii)$ follow. ∎

Figure 10: Polynomials of degree $\leq 4$ for $X=1+2i$

###  6.5 Polynomial rings in one generator over ${{\mathbb S}}[\mu_{6,+}]$

######  Proposition 6.13.

$(i)$ The number $X=2-j$ is an ${{\mathbb S}}[\mu_{6,+}]$-generator of the
ring $R={\mathbb Z}(j)$.
$(ii)$ The hold is given by $h(j)=j+1$, $h(j^{2})=j^{2}+1$, $h(0)=1$ and

$h(1)=X+j,\ \ h(-j^{2})=-j^{2}\,X+j^{2},\ \ h(-j)=-1+X$

$(iii)$ The field $R_{1}$ is the finite field ${\mathbb F}_{7}$.
$(iv)$ The projective limit $\varprojlim R_{m}$ is the Witt ring $W({\mathbb
F}_{7})={\mathbb Z}_{7}$ and the ring $R_{m}$ is the quotient of $W({\mathbb
F}_{7})$ by $7^{m}\,W({\mathbb F}_{7})$.

The proof can be easily deduced from [28], §4.6.

Figure 11: Polynomials of degree $\leq 2$ for
$X=\frac{5}{2}-\frac{i\sqrt{3}}{2}$

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Alain Connes

Collège de France

3 Rue d’Ulm

F-75005 Paris, France

IHES

35 Rte de Chartres

91440 Bures-sur-Yvette, France

Email<EMAIL_ADDRESS>
Caterina Consani

Department of Mathematics

The Johns Hopkins University

3400 N Charles Street

Baltimore MD 21218, USA

Email<EMAIL_ADDRESS>