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\title{Number-theory exercises, II.III}
\author{David Pierce}
\date{\today}

\address{Mathematics Dept\\
Middle East Tech.\ Univ.\\
%Middle East Technical University\\
Ankara 06531, Turkey}

\email{dpierce@metu.edu.tr}
\urladdr{http://www.math.metu.edu.tr/~dpierce/courses/366/}


\usepackage{amssymb,amsmath,amsthm}
%\usepackage{amscd}     % commutative diagram
%\usepackage[mathscr]{euscript}
\usepackage{url}
\usepackage{verbatim}  % allows a comment environment:

%\usepackage{parskip}   % paragraphs not indented, but separated by spaces

\usepackage{hfoldsty}  % gives old-style numerals


\renewcommand{\theenumi}{\alph{enumi}}
\renewcommand{\labelenumi}{\textnormal{(\theenumi)}}

\renewcommand{\theenumii}{\roman{enumii}}
\renewcommand{\labelenumii}{\textnormal{(\theenumii)}}

\newcommand{\defn}[2]{\textbf{#1#2}}

\newcommand{\norm}[1]{\operatorname{N}(#1)}
\newcommand{\size}[1]{\lvert#1\rvert}

\newcommand{\stnd}[1]{\mathbb{#1}}
\newcommand{\N}{\stnd{N}}         % natural numbers
%\newcommand{\varN}{\omega}        % my usual preference for this
%\newcommand{\Zp}{\Z_{+}}          % positive integers
\newcommand{\Z}{\stnd{Z}}         % integers
\newcommand{\Q}{\stnd{Q}}         % rationals
%\newcommand{\Pri}{\stnd{P}}         % primes
\newcommand{\R}{\stnd{R}}         % reals
\newcommand{\C}{\stnd{C}}         % complex numbers

\newcommand{\mi}{\mathrm i}
\newcommand{\gi}{\Z[\mi]}

\let\oldsqrt\sqrt
\renewcommand{\sqrt}[2][1]{\oldsqrt{\vphantom{#1}}#2}
\newcommand{\lat}[1][\alpha,\beta]{\langle#1\rangle} % lattice
\newcommand{\included}{\subseteq}      % [the name suggests the meaning here]
\renewcommand{\setminus}{\smallsetminus}
\newcommand{\ord}[1][\Lambda]{\mathfrak O_{#1}}
\newcommand{\roi}[1][K]{\mathfrak O_{#1}}  % ring of integers

%\input{../Notes/abbrevs}

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%\newtheorem{theorem}{Theorem}
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%\newtheorem{definition}[theorem]{Definition}
%\newtheorem{parag}[theorem]{}
%\newtheorem{ex}[theorem]{Exercise}
\newtheorem{xca}{Exercise}

\theoremstyle{remark}

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%\newcommand{\rmk}[1]{\marginpar[\flushright#1]{\flushleft#1}}

\begin{document}
  \maketitle\thispagestyle{empty}

  \begin{xca}
    Verify that the integers of a quadratic field do compose a ring.
  \end{xca}
\vfill
  \begin{xca}
    Suppose $\tau=(15+3\sqrt{17})/4$.  Find $A$, $B$, and $C$ in $\Z$
    such that $A\tau^2+B\tau+C=0$ and $\gcd(A,B,C)=1$.
  \end{xca}
\vfill
  \begin{xca}
    Suppose $A\tau^2+B\tau+C=0$ for some $A$, $B$, and $C$ in $\Z$,
    where $A>0$ and $\gcd(A,B,C)=1$.
    \begin{enumerate}
      \item\label{item:a}
Show $\lat[1,A\bar\tau]\lat[1,\tau]=\lat[1,\tau]$.
\item\label{item:b}
Show $\lat[A,A\bar\tau]\lat[1,\tau]=\lat[1,A\bar\tau]$.
\item
Using~\eqref{item:a} and~\eqref{item:b}, show
$\ord=\lat[1,A\bar\tau]$, where $\Lambda=\lat[1,\tau]$. 
    \end{enumerate}
  \end{xca}
\vfill
  \begin{xca}
    Let $\Lambda$ be the lattice
    \begin{equation*}
    \Bigl\langle\frac{3+5\sqrt 6}2,\frac{6+\sqrt 6}3\Bigr\rangle
    \end{equation*}
    of $\Q(\sqrt 6)$.  Find $\ord$. 
  \end{xca}
\vfill
  \begin{xca}
    Suppose $\tau\in\C\setminus\Q$.  Show that the following are equivalent:
    \begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}
      \item
$A\tau^2+B\tau+C=0$ for some $A$, $B$, and $C$ in $\Z$;
\item
$\alpha\lat[1,\tau]\included\lat[1,\tau]$ for some $\alpha$ in $\C\setminus\Z$.
    \end{enumerate}
  \end{xca}
\vfill
  \begin{xca}
    Let $f(x,y)$ be the quadratic form
    \begin{equation*}
      60x^2+224xy-735y^2.
    \end{equation*}
    \begin{enumerate}
      \item
Find the discriminant of $f$ in the form $n\sqrt d$, where $n$ and $d$
are rational integers, and $d$ is square-free.
\item
Find all solutions from $\Z$ of $f(x,y)=1$.
\item
Find all solutions from $\Z$ of $f(x,y)=6$.
    \end{enumerate}
  \end{xca}
\vfill
  \begin{xca}
    For every lattice $\Lambda$ of a quadratic field $K$, show that
    the units of $\ord$ are just the units of $\roi$ that are in
    $\ord$.  
  \end{xca}

\vfill
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\end{document}

