Representation of $\overline{\mathbb{Q}}$ in One Dimension

Question

Introduction: Below gives an approach to realize the algebraic numbers $\overline{\mathbb{Q}}\subseteq\mathbb{C}$ within a $1$-dimensional compact connected abelian group (solenoid). Essentially a trade is made: sacrifice path-connectedness in $\mathbb{C}$ to exhibit an algebraic, topological, analytic, and geometric setting in $1$ real dimension for $\overline{\mathbb{Q}}$. The asserted correspondence identifies each algebraic number with a path component of the solenoid.

Let $\boldsymbol{1}=(1+2\mathbb{Z},1+3\mathbb{Z},1+5\mathbb{Z},\dots)\in\Delta\,\colon\!= \prod_{p\in\mathbb{P}}\frac{\mathbb{Z}}{p\mathbb{Z}}$, a commutative profinite ring with identity. Identify $\mathbb{Z}$ with the dense subgroup $\mathbb{Z}\boldsymbol{1}\subseteq\Delta$. A $\widehat{\mathbb{Z}}$-module structure is defined on $\Delta$ by continuously extending the natural multiplication $\mathbb{Z}\times\Delta\rightarrow\Delta$. The resulting scalar multiplication is compatible with the continuous componentwise ring multiplication on $\Delta$. In other words, $\Delta$ is a finitely generated $\widehat{\mathbb{Z}}$-algebra.

Identify $\Delta$ with its topologically isomorphic image in $G\,\colon\!=\frac{\Delta\times\mathbb{R}}{\mathbb{Z}(\boldsymbol{1},1)}$, a $1$-dimensional compact connected abelian group, or solenoid, with a metric topology and (WLOG) total Haar measure $1$.

Define $\mathbb{Q}\Delta$ to be the subgroup of $G$ generated by its profinite subgroups. It is known that

• $\mathbb{Q}\Delta$ is the union of all subgroups $D$ of $G$ containing $\Delta$ for which $[D\,\colon\Delta]<\infty$.
• For each $\boldsymbol{\gamma}\in\mathbb{Q}\Delta$ there is $0\neq n_{\boldsymbol{\gamma}}\in\mathbb{Z}$ with $n_{\boldsymbol{\gamma}}\boldsymbol{\gamma}\in\Delta$.
• $\mathbb{Q}\Delta$ is a $0$-dimensional, non-locally-compact, divisible, incomplete metric subgroup of $G$.
• $G$ has a dense subgroup $X\cong\sum\limits_{p\in\mathbb{P}}\frac{1}{p}\mathbb{Z}$ algebraically isomorphic to the Pontryagin dual of $G$.
• $G$ is topologically isomorphic to $\frac{\mathbb{Q}\Delta\times\mathbb{R}}{X(\boldsymbol{1},1)}$ with identfications $\Delta\subseteq\mathbb{Q}\Delta\subseteq G$ and $X\subseteq\mathbb{Q}\Delta\subseteq G$, subject to the caveat that under the identifications the algebro-topological realizations of $\mathbb{Q}\Delta$ and $X$ go from locally compact outside of $G$ to non-locally-compact as subgroups of $G$.

Each algebraic number $\alpha\notin\mathbb{Z}$ has the form $\beta_\alpha/n_\alpha$ for some algebraic integer $\beta_\alpha$ and some minimal positive integer $n_\alpha >1$. Let $s_\alpha(x)$ denote a nonconstant monic irreducible polynomial in $\mathbb{Z}[x]$ with $s_\alpha(\beta_\alpha)=0$.

Let $\mathbb{P}$ denote the set of prime numbers. For each $m\in\mathbb{Z}$, let $\mathbb{P}_{\alpha,m} =\{p_{\alpha,m,1},\dots,p_{\alpha,m,k_m}\}$ according to the prime factorization $s_\alpha(m) = \pm p_{\alpha,m,1}^{r_{\alpha,m,1}}\cdots p_{\alpha,m,k_m}^{r_{\alpha,m,k_m}}$. For example, $\varnothing\neq\mathbb{P}_{\alpha,0}=\{ p_{\alpha,0,1},\dots, p_{\alpha,0,k_0}\}$ because $s_\alpha(0)=\pm p_{\alpha,0,1}^{r_{\alpha,0,1}}\cdots p_{\alpha,0,k_0}^{r_{\alpha,0,k_0}}$ is the constant term of the irreducible polynomial $s_\alpha (x)\in\mathbb{Z}[x]$, and for $q\in\mathbb{P}$ we have $\mathbb{P}_{\alpha,q}=\{ p_{\alpha,q,1},\dots, p_{\alpha,q,k_q}\}$ where $s_\alpha(q) = \pm p_{\alpha,q,1}^{r_{\alpha,q,1}}\cdots p_{\alpha,q,k_q}^{r_{\alpha,q,k_q}}$.

For $p\in\mathbb{P}$, let $Z_{\alpha,\,p}=\{m\in\mathbb{Z}\,\colon p\mid s_\alpha(m)\}$. Let $\overline{Z}_{\alpha,\,p}=\{m+p\mathbb{Z}\in\mathbb{Z}/p\mathbb{Z}\,\colon m\in Z_{\alpha,\,p}\}$. Then

1. $\mathbb{P}_\alpha\,\colon\!= \{p\in\mathbb{P}\,\colon\, p\mid s_\alpha(m)$ for some $m\in\mathbb{Z}\}=\bigcup\limits_{m\in\mathbb{Z}}\mathbb{P}_{\alpha,m}$ is infinite,
2. $Z_{\alpha,\,p}=\varnothing$ if $p\notin\mathbb{P}_\alpha$,
3. $Z_{\alpha,\,p}$ is infinite if $p\in\mathbb{P}_\alpha$,
4. $\lvert\overline{Z}_{\alpha,\,p}\rvert >1$ for infinitely many $p\in\mathbb{P}_\alpha$,
5. $0\in Z_{\alpha,\,p_{0,1}}\cap\cdots\cap Z_{\alpha,\,p_{0,k_0}}$,
6. $\{p\in\mathbb{P}\,\colon Z_{\alpha,\,p}\cap \mathbb{P}_{\alpha,\,p}\neq\varnothing\}=\{ p_{0,1},\dots, p_{0,k_0}\}$.

Define $\Delta_\alpha =\{\boldsymbol{\gamma}\in\Delta\,\colon\, \forall\,p\notin\mathbb{P}_{\alpha,0},\, p\nmid\gamma_p\Leftrightarrow \gamma_p\in Z_{\alpha,\,p}\}\subseteq\Delta$. By (1) and (4), $\Delta_\alpha$ is uncountable. Let $\boldsymbol{\beta_\alpha}$ be the unique element of $\Delta_\alpha$ satisfying $0\le\beta_{\alpha,\,p}<p$ with $\beta_{\alpha,\,p}$ minimal for all $p\in\mathbb{P}$. Define $\boldsymbol{\beta_\alpha}/n_\alpha\in\mathbb{Q}\Delta$ to be the element $\boldsymbol{r_\alpha}=(r_{\alpha,2}+2^{\ell_2}\mathbb{Z},r_{\alpha,3}+3^{\ell_3}\mathbb{Z},r_{\alpha,5}+5^{\ell_5}\mathbb{Z},\dots)$ with $n_\alpha \boldsymbol{r_\alpha}=\boldsymbol{\beta_\alpha}$ and $0\le r_{\alpha,\,p}<p^{\ell_p}$ with $r_{\alpha,\,p}$ minimal for all $p\in\mathbb{P}$.

Define $\sim$ on $\mathbb{Q}\Delta$ by $\boldsymbol{\gamma}\sim\boldsymbol{\delta}\Leftrightarrow$ there exists $\alpha\in\overline{\mathbb{Q}}$ such that $p\mid n_\alpha(\gamma_p-\delta_p)$ for almost all $p\in \mathbb{P}_\alpha$. Write $[\boldsymbol{\gamma}]$ for the equivalence class of $\boldsymbol{\gamma}$.

Define $f\colon\overline{\mathbb{Q}}\rightarrow\mathbb{Q}\Delta/\!\!\sim$ by $f(\alpha)=[\boldsymbol{\beta_\alpha}/n_\alpha]$ for $\alpha=\beta_\alpha/n_\alpha\in\overline{\mathbb{Q}}$ where $n_\alpha=1\Leftrightarrow \beta_\alpha\in\mathbb{Z}$. Then $f$ is well-defined and surjective.

${\rm Aut}\,\mathbb{Z}[x]=\{g\in {\rm End}\,\mathbb{Z}[x]: g(1)=1$ and $g(x)= m\pm x$ for some $m\in\mathbb{Z}\}$ acts on $\mathbb{Z}[x]$, taking monic irreducible polynomials to monic irreducible polynomials, and so defines a group action on $\overline{\mathbb{Q}}$ which induces a well-defined group action on $\mathbb{Q}\Delta/\!\!\sim$ by $t(x)f(\alpha)\,\colon\!= f(t(x)\alpha)$.

Denote the quotient $\overline{\mathbb{Q}\Delta}\,\,\colon\! = {\rm Aut}\,\mathbb{Z}[x]\backslash \mathbb{Q}\Delta/\!\!\sim$. The elements of $\overline{\mathbb{Q}\Delta}$ are the orbits of $[\boldsymbol{\beta_\alpha}/n_\alpha]$ under the induced group action, and these orbits correspond bijectively with the path components of $G$ via ${\rm Aut}\,\mathbb{Z}[x]\backslash [\boldsymbol{\beta_\alpha}/n_\alpha] \leftrightarrow \boldsymbol{\beta_\alpha}/n_\alpha + G_a$, where $G_a$ denotes the path component of $0\in G$.

Thus, $\overline{\mathbb{Q}}\cong \overline{\mathbb{Q}\Delta}$ exhibits a one-to-one correspondence between algebraic numbers and path components of the $1$-dimensional compact connected abelian group $G$.

Question: Is this construction correct? If not, is it fixable? If fixable, what changes are needed?

Answer 1

This construction is essentially correct. Lots of gaps have to be filled in. The argument needs to identify how distinct zeros of the same monic irrreducible polynomial are differentiated by the construction, even if it is just an existence argument. The various claims about group actions must be verified (e.g., well-defined). The final composite map must be shown bijective and a homomorphism. The objects asserted to be path components must be proven to be, and the bijection with the collection of all path components needs to be verified. The realization of $\mathbb{Q}\Delta$ inside $G$ as the product of all Prüfer groups should be accompanied by descriptions/comparisons of all the intervening topologies: subspace topology, topology induced by completing the topology induced by a non-Archimedean metric on the (dense) torsion subgroup, and the final topology coherent with countable ascending chain of profinite subgroups each of finite index in the next. Note that $\mathbb{Q}\Delta$ is Borel-measurable as a union of closed sets (facilitating Haar integration over).

Add examples to illustrate why this seemingly pathological construct is worth the trouble. Give an example of a process or construction which can be done more simply in the $1$-dimensional setting of $G$ versus the $2$-dimensional setting of $\mathbb{C}$. Give examples illustrating how Haar integration in $G$ parallels computations with Lebesgue integration in $\mathbb{C}$.