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/old/compute_degree.tex | 320 +++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 320 insertions(+) create mode 100644 tex/old/compute_degree.tex (limited to 'tex/old/compute_degree.tex') diff --git a/tex/old/compute_degree.tex b/tex/old/compute_degree.tex new file mode 100644 index 0000000..836a229 --- /dev/null +++ b/tex/old/compute_degree.tex @@ -0,0 +1,320 @@ +\documentclass[10pt,a4paper]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage[all]{xy} +\usepackage{amsfonts} +\usepackage{color} +\usepackage{amssymb} +\usepackage{float} +\usepackage[a4paper, top=3cm, bottom=3cm, left=2.5cm, right=2.5cm]{geometry} + +\DeclareMathOperator{\alg}{alg} +\DeclareMathOperator{\obj}{Obj} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\hol}{Hol} +\DeclareMathOperator{\aut}{Aut} +\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} + + +\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 -- cgit v1.3