From 1437da00eeb4a7581df1b54ffc7680ffb491d690 Mon Sep 17 00:00:00 2001 From: Sebastiano Tronto Date: Mon, 15 Feb 2021 11:30:51 +0100 Subject: Fixed for python3 --- tex/README | 7 + tex/af_code.aux | 11 + tex/af_code.log | 764 ++++++++++++++++++++++++++++++++++++++ tex/af_code.pdf | Bin 0 -> 212906 bytes tex/af_code.synctex.gz | Bin 0 -> 37116 bytes tex/af_code.tex | 474 +++++++++++++++++++++++ tex/old/compute_degree.aux | 14 + tex/old/compute_degree.log | 706 +++++++++++++++++++++++++++++++++++ tex/old/compute_degree.pdf | Bin 0 -> 176578 bytes tex/old/compute_degree.synctex.gz | Bin 0 -> 44626 bytes tex/old/compute_degree.tex | 320 ++++++++++++++++ tex/preprint.pdf | Bin 0 -> 293067 bytes 12 files changed, 2296 insertions(+) create mode 100644 tex/README create mode 100644 tex/af_code.aux create mode 100644 tex/af_code.log create mode 100644 tex/af_code.pdf create mode 100644 tex/af_code.synctex.gz create mode 100644 tex/af_code.tex create mode 100644 tex/old/compute_degree.aux create mode 100644 tex/old/compute_degree.log create mode 100644 tex/old/compute_degree.pdf create mode 100644 tex/old/compute_degree.synctex.gz create mode 100644 tex/old/compute_degree.tex create mode 100644 tex/preprint.pdf (limited to 'tex') 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 +LaTeX2e <2017-04-15> +Babel <3.18> and hyphenation patterns for 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+\DeclareMathOperator{\gal}{Gal} +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\res}{res} +\DeclareMathOperator{\im}{Im} +\DeclareMathOperator{\Id}{Id} +\DeclareMathOperator{\fib}{Fib} +\DeclareMathOperator{\spec}{Spec} +\DeclareMathOperator{\proj}{Proj} +\DeclareMathOperator{\trdeg}{trdeg} +\DeclareMathOperator{\car}{char} +\DeclareMathOperator{\Frac}{Frac} +\DeclareMathOperator{\reduced}{red} +\DeclareMathOperator{\real}{Re} +\DeclareMathOperator{\imag}{Im} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\den}{den} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\lcm}{lcm} +\DeclareMathOperator{\rad}{rad} +\DeclareMathOperator{\ord}{ord} +\DeclareMathOperator{\Br}{Br} +\DeclareMathOperator{\inv}{inv} +\DeclareMathOperator{\Nm}{Nm} +\DeclareMathOperator{\Tr}{Tr} +\DeclareMathOperator{\an}{an} +\DeclareMathOperator{\op}{op} +\DeclareMathOperator{\sep}{sep} +\DeclareMathOperator{\unr}{unr} +\DeclareMathOperator{\et}{\acute et} +\DeclareMathOperator{\ev}{ev} +\DeclareMathOperator{\gl}{GL} +\DeclareMathOperator{\SL}{SL} +\DeclareMathOperator{\mat}{Mat} +\DeclareMathOperator{\ab}{ab} +\DeclareMathOperator{\tors}{tors} +\DeclareMathOperator{\ed}{ed} + +\newcommand{\grp}{\textsc{Grp}} +\newcommand{\set}{\textsc{Set}} +\newcommand{\x}{\mathbf{x}} +\newcommand{\naturalto}{\overset{.}{\to}} +\newcommand{\qbar}{\overline{\mathbb{Q}}} +\newcommand{\zbar}{\overline{\mathbb{Z}}} + +\newcommand{\pro}{\mathbb{P}} +\newcommand{\aff}{\mathbb{A}} +\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} +\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{$n-11$} +\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{$q0}\,|\sqrt{s}\in \mathbb{Q}_x\right\}\,|\,s\in S\right\}$ +\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{$n-10}\,|\sqrt{s}\in \mathbb{Q}_x\right\}\,|\,s\in S\right\}$ +\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}} 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+\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} \ No newline at end of file diff --git a/tex/preprint.pdf b/tex/preprint.pdf new file mode 100644 index 0000000..061fd7c Binary files /dev/null and b/tex/preprint.pdf differ -- cgit v1.3