Usbrth

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

Let $boldsymbol1=(1+2mathbbZ,1+3mathbbZ,1+5mathbbZ,dots)inDelta,colon!= prod_pinmathbbPfracmathbbZpmathbbZ$, a commutative profinite ring with identity. Identify $mathbbZ$ with the dense subgroup $mathbbZboldsymbol1subseteqDelta$. A $widehatmathbbZ$-module structure is defined on $Delta$ by continuously extending the natural multiplication $mathbbZtimesDeltarightarrowDelta$. The resulting scalar multiplication is compatible with the continuous componentwise ring multiplication on $Delta$. In other words, $Delta$ is a finitely generated $widehatmathbbZ$-algebra.

Identify $Delta$ with its topologically isomorphic image in $G,colon!=fracDeltatimesmathbbRmathbbZ(boldsymbol1,1)$, a $1$-dimensional compact connected abelian group, or solenoid, with a metric topology and (WLOG) total Haar measure $1$.

Define $mathbbQDelta$ to be the subgroup of $G$ generated by its profinite subgroups. It is known that

• 
  $mathbbQDelta$ is the union of all subgroups $D$ of $G$ containing $Delta$ for which $[D,colonDelta]<infty$.


• For each $boldsymbolgammainmathbbQDelta$ there is $0neq n_boldsymbolgammainmathbbZ$ with $n_boldsymbolgammaboldsymbolgammainDelta$.


• 
  $mathbbQDelta$ is a $0$-dimensional, non-locally-compact, divisible, incomplete metric subgroup of $G$.


• 
  $G$ has a dense subgroup $Xcongsumlimits_pinmathbbPfrac1pmathbbZ$ algebraically isomorphic to the Pontryagin dual of $G$.


• 
  $G$ is topologically isomorphic to $fracmathbbQDeltatimesmathbbRX(boldsymbol1,1)$ with identfications $DeltasubseteqmathbbQDeltasubseteq G$ and $XsubseteqmathbbQDeltasubseteq G$, subject to the caveat that under the identifications the algebro-topological realizations of $mathbbQDelta$ and $X$ go from locally compact outside of $G$ to non-locally-compact as subgroups of $G$.

Each algebraic number $alphanotinmathbbZ$ 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 $mathbbZ[x]$ with $s_alpha(beta_alpha)=0$.

Let $mathbbP$ denote the set of prime numbers. For each $minmathbbZ$, let $mathbbP_alpha,m =p_alpha,m,1,dots,p_alpha,m,k_m$ according to the prime factorization $s_alpha(m) = pm p_alpha,m,1^r_alpha,m,1cdots p_alpha,m,k_m^r_alpha,m,k_m$. For example, $varnothingneqmathbbP_alpha,0= p_alpha,0,1,dots, p_alpha,0,k_0$ because $s_alpha(0)=pm p_alpha,0,1^r_alpha,0,1cdots p_alpha,0,k_0^r_alpha,0,k_0$ is the constant term of the irreducible polynomial $s_alpha (x)inmathbbZ[x]$, and for $qinmathbbP$ we have $mathbbP_alpha,q= p_alpha,q,1,dots, p_alpha,q,k_q$ where $s_alpha(q) = pm p_alpha,q,1^r_alpha,q,1cdots p_alpha,q,k_q^r_alpha,q,k_q$.

For $pinmathbbP$, let $Z_alpha,,p=minmathbbZ,colon pmid s_alpha(m)$. Let $overlineZ_alpha,,p=m+pmathbbZinmathbbZ/pmathbbZ,colon min Z_alpha,,p$. Then

1. 
   $mathbbP_alpha,colon!= pinmathbbP,colon, pmid s_alpha(m)$ for some $minmathbbZ=bigcuplimits_minmathbbZmathbbP_alpha,m$ is infinite,


2. 
   $Z_alpha,,p=varnothing$ if $pnotinmathbbP_alpha$,


3. 
   $Z_alpha,,p$ is infinite if $pinmathbbP_alpha$,


4. 
   $lvertoverlineZ_alpha,,prvert >1$ for infinitely many $pinmathbbP_alpha$,


5. 
   $0in Z_alpha,,p_0,1capcdotscap Z_alpha,,p_0,k_0$,


6. 
   $pinmathbbP,colon Z_alpha,,pcap mathbbP_alpha,,pneqvarnothing= p_0,1,dots, p_0,k_0$.

Define $Delta_alpha =boldsymbolgammainDelta,colon, forall,pnotinmathbbP_alpha,0,, pnmidgamma_pLeftrightarrow gamma_pin Z_alpha,,psubseteqDelta$. By (1) and (4), $Delta_alpha$ is uncountable. Let $boldsymbolbeta_alpha$ be the unique element of $Delta_alpha$ satisfying $0lebeta_alpha,,p<p$ with $beta_alpha,,p$ minimal for all $pinmathbbP$. Define $boldsymbolbeta_alpha/n_alphainmathbbQDelta$ to be the element $boldsymbolr_alpha=(r_alpha,2+2^ell_2mathbbZ,r_alpha,3+3^ell_3mathbbZ,r_alpha,5+5^ell_5mathbbZ,dots)$ with $n_alpha boldsymbolr_alpha=boldsymbolbeta_alpha$ and $0le r_alpha,,p<p^ell_p$ with $r_alpha,,p$ minimal for all $pinmathbbP$.

Define $sim$ on $mathbbQDelta$ by $boldsymbolgammasimboldsymboldeltaLeftrightarrow$ there exists $alphainoverlinemathbbQ$ such that $pmid n_alpha(gamma_p-delta_p)$ for almost all $pin mathbbP_alpha$. Write $[boldsymbolgamma]$ for the equivalence class of $boldsymbolgamma$.

Define $fcolonoverlinemathbbQrightarrowmathbbQDelta/!!sim$ by $f(alpha)=[boldsymbolbeta_alpha/n_alpha]$ for $alpha=beta_alpha/n_alphainoverlinemathbbQ$ where $n_alpha=1Leftrightarrow beta_alphainmathbbZ$. Then $f$ is well-defined and surjective.

$rm Aut,mathbbZ[x]=gin rm End,mathbbZ[x]: g(1)=1$ and $g(x)= mpm x$ for some $minmathbbZ$ acts on $mathbbZ[x]$, taking monic irreducible polynomials to monic irreducible polynomials, and so defines a group action on $overlinemathbbQ$ which induces a well-defined group action on $mathbbQDelta/!!sim$ by $t(x)f(alpha),colon!= f(t(x)alpha)$.

Denote the quotient $overlinemathbbQDelta,,colon! = rm Aut,mathbbZ[x]backslash mathbbQDelta/!!sim$. The elements of $overlinemathbbQDelta$ are the orbits of $[boldsymbolbeta_alpha/n_alpha]$ under the induced group action, and these orbits correspond bijectively with the path components of $G$ via $rm Aut,mathbbZ[x]backslash [boldsymbolbeta_alpha/n_alpha] leftrightarrow boldsymbolbeta_alpha/n_alpha + G_a$, where $G_a$ denotes the path component of $0in G$.

Thus, $overlinemathbbQcong overlinemathbbQDelta$ 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?