Fixed for python3

12 files changed, 2296 insertions, 0 deletions

diff --git a/tex/README b/tex/README
new file mode 100644
index 0000000..0cfde24
--- /dev/null
+++ b/tex/README
@@ -0,0 +1,7 @@
+Here we collect some incomplete and work-in-progress documentation.
+
+In compute_degrees we prove that our way to compute the degrees gives the
+correct result.
+
+In af_code we break down the code for the "adelic failure" part of the script
+and we check that it computes the degrees as described in compute_degrees.
diff --git a/tex/af_code.aux b/tex/af_code.aux
new file mode 100644
index 0000000..1f8fc85
--- /dev/null
+++ b/tex/af_code.aux
@@ -0,0 +1,11 @@
+\relax 
+\citation{PST1}
+\citation{PST1}
+\citation{DebryPerucca}
+\@writefile{loa}{\contentsline {algorithm}{\numberline {1}{\ignorespaces Compute the adelic failure}}{2}}
+\@writefile{loa}{\contentsline {algorithm}{\numberline {2}{\ignorespaces Adelic failure, case $G\leq \mathbb  {Q}^\times $}}{4}}
+\@writefile{loa}{\contentsline {algorithm}{\numberline {3}{\ignorespaces Adelic failure, case $d\not =-1$, $n\leq d$}}{5}}
+\@writefile{loa}{\contentsline {algorithm}{\numberline {4}{\ignorespaces Adelic failure, case $d\not =-1$, $n\geq d+2$}}{6}}
+\@writefile{loa}{\contentsline {algorithm}{\numberline {5}{\ignorespaces Adelic failure, case $d\not =-1$, $n= d+1$}}{7}}
+\bibcite{DebryPerucca}{1}
+\bibcite{PST1}{2}
diff --git a/tex/af_code.log b/tex/af_code.log
new file mode 100644
index 0000000..126f78c
--- /dev/null
+++ b/tex/af_code.log
@@ -0,0 +1,764 @@
+This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Debian) (preloaded format=pdflatex 2019.9.20)  20 SEP 2019 10:04
+entering extended mode
+ \write18 enabled.
+ %&-line parsing enabled.
+**af_code.tex
+(./af_code.tex
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+\usepackage{amssymb}
+\usepackage{float}
+\usepackage[a4paper, top=3cm, bottom=3cm, left=2.5cm, right=2.5cm]{geometry}
+\usepackage{listings}
+\usepackage{algorithm}
+%\usepackage{algorithmic}
+\usepackage{algpseudocode}
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+\newtheorem{lemma}{Lemma}
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+\newtheorem{corollary}[lemma]{Corollary}
+\newtheorem{definition}[lemma]{Definition}
+\newtheorem{theorem}[lemma]{Theorem}
+\newtheorem{cond-thm}[lemma]{Conditional Theorem}
+\theoremstyle{definition}
+\newtheorem{remark}[lemma]{Remark}
+
+%\author{Sebastiano Tronto}
+\title{Computation of the adelic failure}
+
+
+\begin{document}
+
+\chapter*{Computation of the adelic failure}
+
+The aim of this document is bridge the gap between the theory developed in \cite{PST1} and the function \texttt{adelic\_failure\_gb} that computes the adelic failure. By reading \cite{PST1}, the pseudo-code in this file and the (commented) SageMath code, one can verify that the script produces the correct results.
+
+We begin by giving the code for the function that computes the adelic failure, both in SageMath and in pseudocode. Then we procede to breaking it down into different subcases.
+
+\section*{The SageMath code}
+
+The function \texttt{adelic\_failure\_gb} takes two parameters as input: a list $B=\{B_0,\dots, B_t\}$ and an integer $d$. Each $B_i$ is itself a list of elements of $G$, and we require the following:
+\begin{itemize}
+\item Each element of $B_i=\{B_{i,0},\dots,B_{i,t_i}\}$ has $2$-divisibility $i$, using the terminology of \cite{DebryPerucca}.
+\item $\mathcal{B}=\bigcup_{i=1}^t B_i$ is a $2$-maximal basis for $G$.
+\item The integer $d$ is either $-1$ or $1\leq d\leq t$. For $i\in\{1,\dots,t\}\setminus\{d\}$ we have $B_i\subseteq \mathbb{Q}_+$. If $d\neq -1$ we have $B_{d,0}<0$ and $B_{d,j}>0$ for $j\neq 0$.
+\end{itemize}
+The output is a list $A=\{A_n\mid n\text{ divides }N_0\}$, indexed by the positive divisors of $N_0$, where $N_0=\max(3,t+1)$ if $d=t$, while $N_0=\max(3,t)$ otherwise. Each $A_n=\{A_{n,0},\dots,A_{n,r_n}\}$ is a list of pairs $A_{n,i}=(d_{n,i},f_{n,i})$. For each $n\mid N_0$ and each $i\leq r_n$, the integer $d_{n,i}$ is a divisor of $M_0=d_{N_0,r_{N_0}}$ and a multiple of $2^i$, and $f_{n,i}$ is the \emph{adelic failure}, that is the degree
+\begin{align*}
+f_{n,i}=\left[\mathbb{Q}_{2^i}\left(G^{1/2^i}\right)\cap \mathbb{Q}_{d_{n,i}}:\mathbb{Q}_{2^i}\right].
+\end{align*}
+
+%\lstset{language=Python}
+%\begin{lstlisting}
+%def adelic_failure_gb( B, d ):
+%
+%    ad_fail = []  # The table to be returned at the end.
+%    
+%    if d == len(B)-1:
+%        N = max(3,len(B)+1)
+%    else:
+%        N = max(3,len(B))
+%
+%    # The shortlist grows at each step, so we build it incrementally.
+%    shortlist = []
+%    # The "special element" is (n,b) = \zeta_{2^n}\sqrt{b}.
+%    special_element = (1,1)
+%
+%    M = 1  # M also grows with n.
+%
+%    for n in range( 1, N+1 ): # Read as: 1 \leq n \leq N
+%
+%        # We add the new elements to the shortlist, modifying M if needed.
+%        # This is not done in case we are in the extra "fake" level.
+%        if n-1 < len(B):
+%            for g in B[n-1]:
+%                if g < 0 and n > 1:
+%                    special_element = ( n+1, abs(g)^(1/(2^(n-1))) )
+%                    M = lcm( M, special_embed( special_element ) )
+%                else:
+%                    b = g^(1/(2^(n-1))) # b is 2-indivisible
+%                    shortlist.append( b )
+%                    M = lcm( M, cyc_embed(b) )
+%
+%        # We add a root of an even power of the negative generator, as soon as
+%        # we are beyond its level.
+%        if d != -1 and n == d+2:        
+%            b = abs(B[d][0])^(1/2^d)
+%            shortlist.append( b ) 
+%            M = lcm( M, cyc_embed(b) )
+%
+%        M = lcm(M,2^n)
+%            
+%        if n <= d:
+%            M = lcm( M, 2^(n+1) )
+%
+%        if n == 1 and d >= 1:
+%            shortlist.append(-1)
+%        if n > 1 and -1 in shortlist:
+%            shortlist.remove(-1)
+%
+%        aux = [] # Next line of ad_fail table
+%
+%        for dM in divisors( M ):
+%            if dM % (2^n) != 0:
+%                continue
+%                
+%            S = [ product(s) for s in subsets( shortlist ) ]
+%            H = [ cyc_embed( s ) for s in S ] 
+%            r = len( [ b for b in H if dM % b == 0 ] )
+%
+%            if n <= d and dM % (2^(n+1)) == 0 and n > 1:
+%                r *= 2
+%
+%            if 8 in H and dM % 8 == 0 and (n >= 3 or (n == 2 and n <= d)):
+%                r = r/2
+%
+%            if special_element != (1,1) and special_element[0] == n+1:
+%                nothing_to_do = False
+%                intersecting_QdM = False
+%                for s in S:
+%                    new_special = ( n+1, special_element[1] * s )
+%                    m = special_embed( new_special )
+%                    if n == 2 and m == 4: # \zeta_8 times 2 times square
+%                        nothing_to_do = True
+%                    if dM % m == 0:
+%                        intersecting_QdM = True
+%                if intersecting_QdM and not nothing_to_do:
+%                    r *= 2
+%
+%            aux.append( (dM,r) )
+%        
+%        ad_fail.append(aux)
+%
+%    return ad_fail
+%\end{lstlisting}
+%
+%We have used the following auxiliary functions:
+%
+%\begin{lstlisting}
+%# Computes the minimal cyclotomic field containing \sqrt(b)
+%def cyc_embed( b ):
+%    m = squarefree_part(b)
+%    if m%4 != 1:
+%        m *= 4
+%    return abs(m)
+%
+%# Computes the minimal cyclotomic field containing \zeta_{2^n}\sqrt(b)
+%def special_embed( (n,b) ):
+%    m = squarefree_part(b)
+%    if n == 3 and m % 2 == 0:
+%        return 4 * cyc_embed(m/2)
+%    else:
+%        return lcm( 2^n, cyc_embed(b) )
+%\end{lstlisting}
+\pagebreak
+
+\section*{The pseudo-code}
+We translate the SageMath code into pseudocode for ease of readability.
+
+\begin{algorithm}
+\caption{Compute the adelic failure}
+\begin{algorithmic}
+\State Let $B$, $t$, $d$ and $N$ as described in the previous section
+\State Let $M\leftarrow1$, $\texttt{special\_element}\leftarrow1$ and $\texttt{shortlist}\leftarrow[\,]$
+
+\State
+
+\For {$n=1$ to $N$}
+\If{$n-1<t$}
+\For{$g\in B_{n-1}$}
+\If{$g<0$ and $n>1$}
+\State $\texttt{special\_element}\leftarrow(n+1,\sqrt[2^{n-1}]{|g|})$
+\State $M\leftarrow\lcm(M,\texttt{special\_embed}(\texttt{special\_element}))$
+\Else 
+\State Add $\sqrt[2^{n-1}]{g}$ to \texttt{shortlist}
+\State $M\leftarrow\lcm(M,\texttt{cyc\_embed}(g))$
+\EndIf
+\EndFor
+\EndIf
+
+\State
+
+\If{$n=d+2$ and $d\neq -1$}
+\State Add $\sqrt[2^{d}]{|B_{d,0}|}$ to \texttt{shortlist}
+\State $M\leftarrow\lcm(M,\texttt{cyc\_embed}(|B_{d,0}|))$
+\EndIf
+
+\State
+\If{$n\leq d$}
+\State $M\leftarrow\lcm(M,2^{n+1})$
+\Else
+\State $M\leftarrow\lcm(M,2^n)$
+\EndIf
+\State
+
+\If{$n=1$ and $d\geq 1$}
+\State Add $-1$ to \texttt{shortlist}
+\EndIf
+\State
+\If{$n>1$}
+\State Remove $-1$ from \texttt{shortlist} (if present)
+\EndIf
+\State
+\algstore{alg1}
+\end{algorithmic}
+\end{algorithm}
+\pagebreak
+
+\begin{algorithm}
+\begin{algorithmic}
+\algrestore{alg1}
+\ForAll{$d_M\in \texttt{divisors}(M)$ such that $2^n\,|\,M$}
+\State $S\leftarrow\left\{\prod_{x\in T}x\,|\,T\subseteq\texttt{shortlist}\right\}$
+\State $H\leftarrow\left\{\min\left\{x\in\mathbb{Z}_{>0}\,|\sqrt{s}\in \mathbb{Q}_x\right\}\,|\,s\in S\right\}$
+\State $r\leftarrow\# \left\{s\in S\,|\, \sqrt{s}\in\mathbb{Q}_{d_M}\right\}$
+\State
+\If{$q<n\leq d$ and $2^{n+1}\,|\,{d_M}$}
+\State $r\leftarrow 2r$
+\EndIf
+\State
+\If{$8\in H$ and $8\,|\,d_M$ and (either $n\geq 3$ or $n=2\leq d$)}
+\State $r\leftarrow r/2$
+\EndIf
+\State
+
+\If{$\texttt{special\_element}=\zeta_{2^{n+1}}\sqrt{b}$ for some $b\in\mathbb{Q}$}
+\State $\texttt{specials}\leftarrow\{\zeta_{2^{n+1}}\sqrt{bs}\,|\,s\in S\}$
+\If{$\exists x\in \texttt{specials}$ such that $x\in\mathbb{Q}_{d_M}$ and $\texttt{special\_embed}(s)\neq 4\,\forall s\in\texttt{specials}$}
+\State $r\leftarrow 2r$
+\EndIf
+\EndIf
+\State
+\State Declare $\left[\mathbb{Q}_{2^n}\left(\sqrt[2^n]{G}\right)\cap \mathbb{Q}_{d_M}:\mathbb{Q}_{2^n}\right]=r$.
+
+\EndFor
+\EndFor
+\end{algorithmic}
+\end{algorithm}
+
+\pagebreak
+
+\section*{Pseudo-code, the sub-cases}
+
+We divide the pseudocode in sub-cases. In each sub-case we apply the trivial simplifications to the pseudo-code above.
+
+\subsection*{Case $G\leq \mathbb{Q}_+^\times$}
+
+\begin{algorithm}
+\caption{Adelic failure, case $G\leq \mathbb{Q}^\times$}
+
+\begin{algorithmic}
+\For {$n=1$ to $N$}
+\For{$g\in B_{n-1}$}
+\State Add $\sqrt[2^{n-1}]{g}$ to \texttt{shortlist}
+\State $M\leftarrow\lcm(M,\texttt{cyc\_embed}(g))$
+\EndFor
+\State
+\State $M\leftarrow\lcm(M,2^n)$
+\State
+\ForAll{$d_M\in \texttt{divisors}(M)$ such that $2^n\,|\,M$}
+\State $S\leftarrow\left\{\prod_{x\in T}x\,|\,T\subseteq\texttt{shortlist}\right\}$
+\State $H\leftarrow\left\{\min\left\{x\in\mathbb{Z}_{>0}\,|\sqrt{s}\in \mathbb{Q}_x\right\}\,|\,s\in S\right\}$
+\State $r\leftarrow\# \left\{s\in S\,|\, \sqrt{s}\in\mathbb{Q}_{d_M}\right\}$
+%\State
+\State Declare $\left[\mathbb{Q}_{2^n}\left(\sqrt[2^n]{G}\right)\cap \mathbb{Q}_{d_M}:\mathbb{Q}_{2^n}\right]=\begin{cases}
+r/2&\text{ if }8\in H\text{ and }n\geq 3,\\
+r&\text{ otherwise}.
+\end{cases}$
+\EndFor
+\EndFor
+\end{algorithmic}
+
+\end{algorithm}
+\pagebreak
+\subsection*{Case $d\neq -1$, $n\leq d$}
+For this and the following cases, we assume we are already inside the main \texttt{for} cycle, since we have particular assumptions on $n$.
+\begin{algorithm}
+\caption{Adelic failure, case $d\neq -1$, $n\leq d$}
+\begin{algorithmic}
+\For{$g\in B_{n-1}$}
+\State Add $\sqrt[2^{n-1}]{g}$ to \texttt{shortlist}
+\State $M\leftarrow\lcm(M,\texttt{cyc\_embed}(g))$
+\EndFor
+\State
+\State $M\leftarrow\lcm(M,2^{n+1})$
+\State
+\If{$n=1$ and $d\geq 1$}
+\State Add $-1$ to \texttt{shortlist}
+\EndIf
+\State
+\If{$n>1$}
+\State Remove $-1$ from \texttt{shortlist} (if present)
+\EndIf
+\State
+\ForAll{$d_M\in \texttt{divisors}(M)$ such that $2^n\,|\,M$}
+\State $S\leftarrow\left\{\prod_{x\in T}x\,|\,T\subseteq\texttt{shortlist}\right\}$
+\State $H\leftarrow\left\{\min\left\{x\in\mathbb{Z}_{>0}\,|\sqrt{s}\in \mathbb{Q}_x\right\}\,|\,s\in S\right\}$
+\State $r\leftarrow\# \left\{s\in S\,|\, \sqrt{s}\in\mathbb{Q}_{d_M}\right\}$
+\State
+\If{$n>1$ and $2^{n+1}\,|\,d_M$}
+\State $r\leftarrow 2r$
+\EndIf
+\State Declare $\left[\mathbb{Q}_{2^n}\left(\sqrt[2^n]{G}\right)\cap \mathbb{Q}_{d_M}:\mathbb{Q}_{2^n}\right]=\begin{cases}
+r/2&\text{ if }8\in H\text{ and }n\geq 3,\\
+r/2&\text{ if }8\in H\text{ and }n=2\text{ and }8\,|\,d_M\\
+r&\text{ otherwise}.
+\end{cases}$
+\EndFor
+\end{algorithmic}
+
+\end{algorithm}
+
+
+\pagebreak
+\subsection*{Case $d\neq -1$, $n\geq d+2$}
+
+\begin{algorithm}
+\caption{Adelic failure, case $d\neq -1$, $n\geq d+2$}
+\begin{algorithmic}
+\If{$n-1<t$}
+\For{$g\in B_{n-1}$}
+\State Add $\sqrt[2^{n-1}]{g}$ to \texttt{shortlist}
+\State $M\leftarrow\lcm(M,\texttt{cyc\_embed}(g))$
+\EndFor
+\EndIf
+\State
+
+\If{$n=d+2$}
+\State Add $\sqrt[2^{d}]{|B_{d,0}|}$ to \texttt{shortlist}
+\State $M\leftarrow\lcm(M,\texttt{cyc\_embed}(|B_{d,0}|))$
+\EndIf
+
+\State
+\State $M\leftarrow\lcm(M,2^{n})$
+\State
+\ForAll{$d_M\in \texttt{divisors}(M)$ such that $2^n\,|\,M$}
+\State $S\leftarrow\left\{\prod_{x\in T}x\,|\,T\subseteq\texttt{shortlist}\right\}$
+\State $H\leftarrow\left\{\min\left\{x\in\mathbb{Z}_{>0}\,|\sqrt{s}\in \mathbb{Q}_x\right\}\,|\,s\in S\right\}$
+\State $r\leftarrow\# \left\{s\in S\,|\, \sqrt{s}\in\mathbb{Q}_{d_M}\right\}$
+\State Declare $\left[\mathbb{Q}_{2^n}\left(\sqrt[2^n]{G}\right)\cap \mathbb{Q}_{d_M}:\mathbb{Q}_{2^n}\right]=\begin{cases}
+r/2&\text{ if }8\in H,\\
+r&\text{ otherwise}.
+\end{cases}$
+\EndFor
+\end{algorithmic}
+
+\end{algorithm}
+
+\pagebreak
+
+\subsection*{Case $d\neq -1$, $n= d+1$}
+
+\begin{algorithm}
+\caption{Adelic failure, case $d\neq -1$, $n= d+1$}
+\begin{algorithmic}
+\For{$g\in B_{n-1}$}
+\If{$g<0$}
+\State $\texttt{special\_element}\leftarrow(n+1,\sqrt[2^{n-1}]{|g|})$
+\State $M\leftarrow\lcm(M,\texttt{special\_embed}(\texttt{special\_element}))$
+\Else 
+\State Add $\sqrt[2^{n-1}]{g}$ to \texttt{shortlist}
+\State $M\leftarrow\lcm(M,\texttt{cyc\_embed}(g))$
+\EndIf
+\EndFor
+\State
+\State $M\leftarrow\lcm(M,2^{n})$
+\State
+\State Remove $-1$ from \texttt{shortlist} (if present)
+
+
+\State
+\ForAll{$d_M\in \texttt{divisors}(M)$ such that $2^n\,|\,M$}
+\State $S\leftarrow\left\{\prod_{x\in T}x\,|\,T\subseteq\texttt{shortlist}\right\}$
+\State $H\leftarrow\left\{\min\left\{x\in\mathbb{Z}_{>0}\,|\sqrt{s}\in \mathbb{Q}_x\right\}\,|\,s\in S\right\}$
+\State $r\leftarrow\# \left\{s\in S\,|\, \sqrt{s}\in\mathbb{Q}_{d_M}\right\}$
+\State
+%\State $\texttt{specials}\leftarrow\{\zeta_{2^{n+1}}\sqrt{bs}\,|\,s\in S\}$
+\If{$\exists x\in \{\zeta_{2^{n+1}}\sqrt{bs}\,|\,s\in S\}\cap\mathbb{Q}_{d_M}$ and $\texttt{special\_embed}(s)\neq 4\,\forall s\in\texttt{specials}$}
+\State $r\leftarrow 2r$
+\EndIf
+\State Declare $\left[\mathbb{Q}_{2^n}\left(\sqrt[2^n]{G}\right)\cap \mathbb{Q}_{d_M}:\mathbb{Q}_{2^n}\right]=\begin{cases}
+r/2&\text{ if }8\in H\text{ and }n\geq 3,\\
+r&\text{ otherwise}.
+\end{cases}$
+\EndFor
+\end{algorithmic}
+
+\end{algorithm}
+
+\begin{thebibliography}{10} \expandafter\ifx\csname url\endcsname\relax   \def\url#1{\texttt{#1}}\fi \expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
+
+\bibitem{DebryPerucca}  
+\textsc{Debry, C. - Perucca, A.}: \emph{Reductions of algebraic integers}, J. Number Theory, {\bf 167} (2016), 259--283.
+
+\bibitem{PST1}
+\textsc{Perucca, A. - Sgobba, P. - Tronto, S.}: \emph{Explicit Kummer Theory for the rational numbers}, preprint.
+
+\end{thebibliography}
+
+\end{document}
\ No newline at end of file
diff --git a/tex/old/compute_degree.aux b/tex/old/compute_degree.aux
new file mode 100644
index 0000000..8be55bd
--- /dev/null
+++ b/tex/old/compute_degree.aux
@@ -0,0 +1,14 @@
+\relax 
+\citation{DebryPerucca}
+\citation{DebryPerucca}
+\newlabel{lemma_zero}{{1}{1}}
+\newlabel{degree}{{1}{1}}
+\@writefile{toc}{\contentsline {section}{\numberline {1}Case $G\leq \mathbb  {Q}_+^\times $}{1}}
+\@writefile{toc}{\contentsline {section}{\numberline {2}General case}{2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}General case, $n=1(\leq d)$}{2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}General case, $n=2\leq d$}{2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}General case, $3\leq n\leq d$}{2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.4}General case, $n\geq d+2$}{3}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {2.5}General case, $n=d+1$}{3}}
+\bibcite{DebryPerucca}{1}
+\bibcite{PeruccaSgobba}{2}
diff --git a/tex/old/compute_degree.log b/tex/old/compute_degree.log
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index 0000000..571b9ec
--- /dev/null
+++ b/tex/old/compute_degree.log
@@ -0,0 +1,706 @@
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+entering extended mode
+ \write18 enabled.
+ %&-line parsing enabled.
+**compute_degree.tex
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+   defining Unicode char U+00E1 (decimal 225)
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+   defining Unicode char U+00E3 (decimal 227)
+   defining Unicode char U+00E4 (decimal 228)
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+   defining Unicode char U+00EB (decimal 235)
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+   defining Unicode char U+00ED (decimal 237)
+   defining Unicode char U+00EE (decimal 238)
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+   defining Unicode char U+00F4 (decimal 244)
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diff --git a/tex/old/compute_degree.pdf b/tex/old/compute_degree.pdf
new file mode 100644
index 0000000..156383c
--- /dev/null
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diff --git a/tex/old/compute_degree.synctex.gz b/tex/old/compute_degree.synctex.gz
new file mode 100644
index 0000000..86cccf0
--- /dev/null
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diff --git a/tex/old/compute_degree.tex b/tex/old/compute_degree.tex
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--- /dev/null
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+\newcommand{\quat}{\mathbb{H}}
+\newcommand{\rea}{\mathbb{R}}
+\newcommand{\kiu}{\mathbb{Q}}
+\newcommand{\F}{\mathbb{F}}
+\newcommand{\zee}{\mathbb{Z}}
+\newcommand{\ow}{\mathcal{O}}
+\newcommand{\mcx}{\mathcal{X}}
+\newcommand{\mcy}{\mathcal{Y}}
+\newcommand{\mcs}{\mathcal{S}}
+\newcommand{\mca}{\mathcal{A}}
+\newcommand{\mcb}{\mathcal{B}}
+\newcommand{\mcf}{\mathcal{F}}
+\newcommand{\mcg}{\mathcal{G}}
+\newcommand{\mct}{\mathcal{T}}
+\newcommand{\mcq}{\mathcal{Q}}
+\newcommand{\mcr}{\mathcal{R}}
+\newcommand{\adl}{\mathbf{A}}
+\newcommand{\mbk}{\mathbf{k}}
+\newcommand{\m}{\mathfrak{m}}
+\newcommand{\p}{\mathfrak{p}}
+
+\newcommand{\kbar}{\overline{K}}
+
+\newtheorem{lemma}{Lemma}
+\newtheorem{proposition}[lemma]{Proposition}
+\newtheorem{conjecture}[lemma]{Conjecture}
+\newtheorem{corollary}[lemma]{Corollary}
+\newtheorem{definition}[lemma]{Definition}
+\newtheorem{theorem}[lemma]{Theorem}
+\newtheorem{cond-thm}[lemma]{Conditional Theorem}
+\theoremstyle{definition}
+\newtheorem{remark}[lemma]{Remark}
+
+\author{Sebastiano Tronto}
+
+
+\begin{document}
+
+\begin{lemma}
+\label{lemma_zero}
+Let $H\leq \mathbb{Q}^\times$ be a finitely generated subgroup. Assume that $H$ does not contain minus a square of $\mathbb{Q}^\times$ or that $m=1$. Then we have
+\begin{align*}
+\left[\mathbb{Q}_{2^m}\left(\sqrt{H}\right):\mathbb{Q}_{2^m}\right]=\begin{cases}
+\#\overline H/2 & \text{ if }m\geq 3\text{ and }\exists b\in H\text{ with }b\equiv\pm2\pmod{\mathbb{Q}^{\times 2}},\\
+\#\overline H&\text{ otherwise}.
+\end{cases}
+\end{align*}
+where $\overline{H}$ is the image of $H\cdot \mathbb{Q}^{\times 2}$ in $\mathbb{Q}^\times/\mathbb{Q}^{\times 2}$.
+\begin{proof}
+Clearly we may assume that $H$ is generated by suqarefree integers $\{g_1,\dots, g_r\}$, where $r=\#\overline H$. In fact, we have that $\mathbb{Q}_{2^m}(\sqrt{H})=\mathbb{Q}_{2^m}(\sqrt{H'})$ for any $H'$ such that $(H\cdot \mathbb{Q}^{\times 2})/\mathbb{Q}^{\times 2}=(H'\cdot \mathbb{Q}^{\times 2})/\mathbb{Q}^{\times 2}$. Recall moreover that by {\color{red}Lemma 13} if there is $\pm2$ times a square in $H$ we can assume that, say, $g_1=\pm 2$.
+
+Assume first that $m\geq 2$, so that $-1\not\in H$ by assumption. In this case we can work over $\mathbb Q_4$ and use Theorem 18 of \cite{DebryPerucca}. We just need to compute the divisibility parameters over $\mathbb{Q}_4$:
+\begin{align*}
+d_1=\begin{cases}
+0&\text{ if }g_1\neq\pm2\\
+1&\text{ if }g_1=\pm2
+\end{cases},
+&&
+d_i=0
+\quad \text{ for $i=2,\dots, r$},\\
+h_1=\begin{cases}
+0&\text{ if } 0\leq g_1\neq2\\
+1&\text{ if } -2\neq g_1<0\\
+2&\text{ if } g_1=\pm 2
+\end{cases}, &&
+h_i=\begin{cases}
+0&\text{ if }g_i>0\\
+1&\text{ if }g_i<0
+\end{cases}
+\quad \text{ for $i=2,\dots, r$}.
+\end{align*}
+Thus, keeping the notation of the aformentioned Theorem, we get
+\begin{align*}
+n_1=\min(1,d_1)=\begin{cases}
+0&\text{ if }g_1\neq\pm2\\
+1&\text{ if }g_1=\pm2
+\end{cases},&& n_i=0\quad \text{ for $i=2,\dots, r$}.
+\end{align*}
+Thus we get
+\begin{align*}
+v_2\left[\mathbb{Q}_{2^m}(\sqrt{H}):\mathbb Q_{2^m}\right]&=\max(h_1+n_1,\dots, h_r+n_r,m)-m+r-\sum_{i=1}^rn_i=\\
+&=\begin{cases}
+\max(3,m)-m+r-\sum_{i=1}^rn_i&\text{ if }\pm2\in H\\
+r-\sum_{i=1}^rn_i&\text{ if }\pm2\not \in H
+\end{cases}\\
+&=\begin{cases}
+1+r-1&\text{ if }m=2\text{ and }\pm2\in H\\
+r-1&\text{ if }m\geq3\text{ and }\pm2\in H\\
+r&\text{ if }\pm2\not\in H
+\end{cases}
+\end{align*}
+which is what we want.
+
+Assume now that $m=1$. If $-1\not\in H$, we get the desired result directly from Lemma 19 of \cite{DebryPerucca} applied with $G=H$, using the computations that we did in the previous case. In case $-1\in H$, let $H'$ be any subgroup of $H$ such that $H=H'\oplus\langle-1\rangle$. Notice that we have $\#\overline {H'}=r-1$, so that Lemma 19 with $G=H'$ again gives our result, and the Proposition is proved.
+\end{proof}
+\end{lemma}
+
+Let $G\leq \mathbb{Q}^\times$ be a finitely generated torsion-free subgroup of rank $r$ and let $M$ and $n$ be integers such that $2^n\,|\,M$. We want to compute the degree
+\begin{align}
+\label{degree}
+\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right].
+\end{align}
+
+We will use the same notation as that of Remark 17 of Pietro's file.
+
+\section{Case $G\leq \mathbb{Q}_+^\times$}
+
+Assume that $G\leq \mathbb{Q}_+^\times$. In this case, by Remark 17, we have that
+\begin{align*}
+\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M =\mathbb{Q}_{2^n}\left(\sqrt{H}\right).
+\end{align*}
+
+Let $\overline{H}$ be the image of $H$ in $\mathbb{Q^\times}/\mathbb{Q}^{\times 2}$. By Remark 17 and Lemma \ref{lemma_zero} above, the degree (\ref{degree}) is given by
+\begin{align*}
+\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]=
+\begin{cases}
+\#\overline H/2 & \text{if }n\geq 3\text{ and }2\in H,\\
+\#\overline H&\text{ otherwise}.
+\end{cases}
+\end{align*}
+
+\section{General case}
+
+Let $\mathcal{B}$ be a basis for $G$ and let $\mathcal{B}_i\subseteq \mathcal{B}$ be the subset of basis elements of $2$-divisibility $i$. Call also $L=\max d_i$ the largest $2$-divisiblity parameter. In this way $\mathcal{B}_0,\dots,\mathcal{B}_L$ is a partition of $\mathcal{B}$.
+
+As explained in ({\color{red}ref}) we may assume that there is at most one negative basis element. Since we have dealt with the $G\subseteq \mathbb{Q}_+$ case in the previous section, we assume that such an element exists and that it has $2$-divisibility $d$. We call this element $g_0$.
+
+It is (or will be?) clear ({\color{red}but we should explain it}) that it actually does not matter if we have negative elements of divisibility $0$: that case is treated exactly as the case $G\subseteq \mathbb{Q}_+$. In conclusion, we assume that:
+\begin{align*}
+\mathcal{B}_1,\dots,\mathcal{B}_{d-1},\mathcal{B}_{d+1},\dots,\mathcal{B}_L\subseteq \mathbb{Q}_+,\\
+g_0<0 \text{ and }\mathcal{B}_d\setminus \{g_0\}\subseteq \mathbb{Q}_+,\\
+d\geq 1.
+\end{align*}
+
+We also let
+\begin{align*}
+N=\begin{cases}
+\max(3,L)&\text{if }d\neq L,\\
+\max(3,L+1)&\text{if }d=L.
+\end{cases}
+\end{align*}
+
+\subsection{General case, $n=1(\leq d)$}
+This case can be treated as follows: let $\mathcal{S}'=\mathcal{S}\cup \{-1\}$ and let $H'$ be constructed from $\mathcal{S}'$ in the exact same way as $H$ is constructed from $\mathcal{S}$. Then it's easy to check ({\color{red}it follows from the ``torsion case'' for $G$, it is for sure in some other file}) that $\mathbb{Q}_{2^n}\left(\sqrt{H'}\right)=\mathbb{Q}_{2^{w'}}\left(\sqrt{H}\right)$, where $w'=\min(v_2(M),n+1)$ (as in Remark 17). Then we can again use Lemma \ref{lemma_zero} and conclude that
+\begin{align*}
+\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]=
+\#\overline{H'},
+\end{align*}
+where $\#\overline{H'}$ is the image of $H'$ in $\mathbb{Q}^\times/\mathbb{Q}^{\times 2}$.
+
+\subsection{General case, $n=2\leq d$}
+We consider two cases:
+\begin{itemize}
+\item If $v_2(M)=2$ we have $\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]=\left[\mathbb{Q}_4\left(\sqrt{H}\right):\mathbb{Q}_4\right]=\#\overline{H}$ by Lemma \ref{lemma_zero}.
+\item If $v_2(M)\geq 3$ we have
+\begin{align*}
+\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]&=\left[\mathbb{Q}_8\left(\sqrt{H}\right):\mathbb{Q}_4\right]=\left[\mathbb{Q}_8\left(\sqrt{H}\right):\mathbb{Q}_8\right]\cdot \left[\mathbb{Q}_8:\mathbb{Q}_4\right]=\\&=2\left[\mathbb{Q}_8\left(\sqrt{H}\right):\mathbb{Q}_8\right],
+\end{align*}
+which, by Lemma \ref{lemma_zero}, is given by $\#\overline{H}$ if $2 \in H$ and by $2\#\overline{H}$ otherwise.
+\end{itemize}
+
+\subsection{General case, $3\leq n\leq d$}
+We consider two cases:
+\begin{itemize}
+\item If $v_2(M)=3$, by lemma \ref{lemma_zero} we have
+\begin{align*}
+\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]=\left[\mathbb{Q}_8\left(\sqrt{H}\right):\mathbb{Q}_8\right]=\begin{cases}
+\#\overline H/2 & \text{ if }\pm 2\in H,\\
+\#\overline H&\text{ otherwise}.
+\end{cases}
+\end{align*}
+\item If $v_2(M)\geq 4$ we have
+\begin{align*}
+\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]&=\left[\mathbb{Q}_{16}\left(\sqrt{H}\right):\mathbb{Q}_8\right]=\left[\mathbb{Q}_{16}\left(\sqrt{H}\right):\mathbb{Q}_{16}\right]\cdot \left[\mathbb{Q}_{16}:\mathbb{Q}_8\right]=\\&=2\left[\mathbb{Q}_{16}\left(\sqrt{H}\right):\mathbb{Q}_{16}\right],
+\end{align*}
+which, by Lemma \ref{lemma_zero}, is given by $\#\overline{H}$ if $2 \in H$ and by $2\#\overline{H}$ otherwise.
+\end{itemize}
+
+\subsection{General case, $n\geq d+2$}
+By the corresponding case in Remark 17, we simply have
+\begin{align*}
+\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]=\begin{cases}
+\#\overline {H'}/2 & \text{ if }\pm 2\in H,\\
+\#\overline {H'}&\text{ otherwise}.
+\end{cases}
+\end{align*}
+where $H'$ is constructed from $\mathcal{S}'=\mathcal{S}\cup\{B_0\}$ and $\overline{H'}$ is the image of $H'$ in $\mathbb{Q}^\times/\mathbb{Q}^{\times 2}$.
+
+\subsection{General case, $n=d+1$}
+We distinguish between some cases.
+\begin{itemize}
+\item Assume $n=2$ (thus $d=3$) and $v_2(g_0)=2$ (i.e. $2$ divides the square-free part of $B_0$, where $g_0=-B_0^{2^d}$). Then we write the square-free part of $B_0$ as $2s$ for some odd square-free $s\in\mathbb{Z}$. Then letting $\mathcal{S}':=\mathcal{S}\cup \{s\}$ and construct $H'$ from $\mathcal{S}'$ in the usual way. By Remark 17 we have
+\begin{align*}
+\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]=\left[\mathbb{Q}_{2^n}\left(\sqrt{H'}\right):\mathbb{Q}_{2^n}\right]=\#\overline{H'}.
+\end{align*}
+But we can be more precise and say that
+\begin{align*}
+\#\overline{H'}=\begin{cases}
+2\#\overline{H}&\text{if }\sqrt{xs}\in\mathbb{Q}_M\text{ for some }x\in\mathcal{S}\text{ and }s\not\in \mathcal{S},\\
+\#\overline{H}&\text{otherwise}.
+\end{cases}
+\end{align*}
+%\item Assume $n=2$, $2^{n+1}\nmid M$ and either $v_2(g_0)>2$ or $g_0$ is odd. Then $\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]=\#\overline H$.
+%\item Assume $n=2$, $2^{n+1}\,|\,M$ and either $v_2(g_0)>2$ or $g_0$ is odd. ({\color{red}TODO})
+\item Assume $n\geq 2$ and $2^{n+1}\nmid M$. Then
+\begin{align*}
+\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]=\begin{cases}
+\#\overline H/2 & \text{ if }\pm 2\in H\text{ and }n\geq 3,\\
+\#\overline H&\text{ otherwise}.
+\end{cases}
+\end{align*}
+\item Assume $n\geq 2$ and $2^{n+1}\,|\,M$. Following the notation of Remark 17, we have
+\begin{align*}
+\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M=\mathbb{Q}_{2^n}\left(\sqrt{\langle H, H'\rangle}\right)
+\end{align*}
+hence
+\begin{align*}
+\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]&=\left[\mathbb{Q}_{2^n}\left(\sqrt{\langle H, H'\rangle}\right):\mathbb{Q}_{2^n}\right]=\\
+&=\left[\mathbb{Q}_{2^n}\left(\sqrt{\langle H, H'\rangle}\right):\mathbb{Q}_{2^n}\left(\sqrt{H}\right)\right]\cdot \left[\mathbb{Q}_{2^n}\left(\sqrt{H}\right):\mathbb{Q}_{2^n}\right].
+\end{align*}
+We claim that
+\begin{align*}
+\left[\mathbb{Q}_{2^n}\left(\sqrt{\langle H, H'\rangle}\right):\mathbb{Q}_{2^n}\left(\sqrt{H}\right)\right]=\begin{cases}
+1&\text{ if }H'=\emptyset\text{ or }H'=\{2\zeta_4\},\\
+2&\text{ otherwise}.
+\end{cases}
+\end{align*}
+To see this, notice that $\sqrt{2\zeta_4}=\zeta_8\sqrt{2}\in\mathbb{Q}_4\subseteq\mathbb{Q}_{2^n}\left(\sqrt{H}\right)$, so the first case is settled. Assume now that there is $x=\zeta_{2^n}b\in H'$ with $x\neq 2\zeta_4$. If $y=\zeta_{2^n}c$ is any other element of $H'$, then we have $\sqrt{x/y}=\sqrt{b/c}$. So if $x,y\in \mathbb{Q}_{2^n}\left(\sqrt{\langle H, H'\rangle}\right)$ we have also $\sqrt{b/c}\in \mathbb{Q}_{2^n}\left(\sqrt{\langle H, H'\rangle}\right)$, which by Kummer theory implies $bc\in H$. But then $y\in \mathbb{Q}_{2^n}\left(\sqrt{H}\right)\left(x\right)$. So we have $\mathbb{Q}_{2^n}\left(\sqrt{\langle H, H'\rangle}\right)=\mathbb{Q}_{2^n}\left(\sqrt{H}\right)(x)$, and the sought degree is $\left[\mathbb{Q}_{2^n}\left(\sqrt{H}\right)(x):\mathbb{Q}_{2^n}\left(\sqrt{H}\right)\right]$, which is in fact $2$ ({\color{red}Do we need to explain this better?}).
+
+We conclude that
+\begin{align*}
+\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]=\begin{cases}
+\#\overline{H}/2&\text{ if } n\geq 3,\,\pm2 \in H\text{ and }H'\subseteq\{2\zeta_4\},\\
+\#\overline{H}&\text{ if }(n<3\text{ or }\pm2\not\in H)\text{ and }H'\subseteq \{2\zeta_4\},\\
+\#\overline{H}&\text{ if } n\geq 3,\,\pm2 \in H\text{ and }H'\not\subseteq\{2\zeta_4\},\\
+2\cdot \#\overline{H}&\text{ if }(n<3\text{ or }\pm2\not\in H)\text{ and }H'\not\subseteq \{2\zeta_4\}.
+\end{cases}
+\end{align*}
+%Let $s$ as in the first subcase of this section and let $\mathcal{C}'$ and $H'$ be as in the last case of Remark 17. We have
+%\begin{align*}
+%\left[\mathbb{Q}_{2^n}\left(G^{1/2^n}\right)\cap \mathbb{Q}_M:\mathbb{Q}_{2^n}\right]=&\left[\mathbb{Q}_{2^n}\left(\sqrt{\langle H,\zeta_{2^n}H'\rangle}\right):\mathbb{Q}_{2^n}\right]=\\
+%=&\left[\mathbb{Q}_{2^n}\left(\sqrt{\langle H,\zeta_{2^n}H'\rangle}\right):\mathbb{Q}_{2^n}\left(\sqrt{H}\right)\right]\cdot \left[\mathbb{Q}_{2^n}\left(\sqrt{H}\right):\mathbb{Q}_{2^n}\right].
+%\end{align*}
+%Notice that, by construction of $H$ and $H'$, the degree $\left[\mathbb{Q}_{2^n}\left(\sqrt{\langle H,\zeta_{2^n}H'\rangle}\right):\mathbb{Q}_{2^n}\left(\sqrt{H}\right)\right]$ is either $1$
+\end{itemize}
+
+\begin{thebibliography}{10} \expandafter\ifx\csname url\endcsname\relax   \def\url#1{\texttt{#1}}\fi \expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
+
+\bibitem{DebryPerucca}  
+\textsc{Debry, C. - Perucca, A.}: \emph{Reductions of algebraic integers}, J. Number Theory, {\bf 167} (2016), 259--283.
+
+\bibitem{PeruccaSgobba}  
+\textsc{Perucca, A. - Sgobba, P.}: \emph{Kummer Theory for Number Fields}, preprint.
+
+\end{thebibliography}
+
+\end{document}
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