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\title{Number-theory exercises, II.V}
\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{\mLambda}{\mathit{\Lambda}}
\newcommand{\mMu}{M}
\newcommand{\ord}[1][\mLambda]{\mathfrak O_{#1}}
\newcommand{\roi}[1][K]{\mathfrak O_{#1}}  % ring of integers
\newcommand{\Fib}[1]{\operatorname{F}_{#1}}

%\input{../Notes/abbrevs}

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%\swapnumbers


%\newtheorem{theorem}{Theorem}
%\newtheorem{axdef}[theorem]{Axiom and definition}
%\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}

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

\theoremstyle{remark}

%\newtheorem{remarks}[theorem]{Remarks}
%\newtheorem{remark}[theorem]{Remark}

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

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

  \begin{xca}
  Prove that the $n$th convergent of $\sqrt 5$ is
  $\displaystyle\frac{2\Fib{3n+2}+\Fib{3n+3}}{\Fib{3n+3}}$.     
  \end{xca}

  \begin{xca}
    Verify that an order $\ord[]$ of $K$ is in particular a lattice
    $\mLambda$ such that $\ord=\ord[]$.
  \end{xca}

  \begin{xca}
      Let $\mLambda$ and $\mMu$ be lattices of $K$.  Prove the
      following. 
  \begin{enumerate}
\item
    $\mLambda+\mMu$ is a lattice, and
    \begin{equation*}
      \lat+\lat[\gamma+\delta]=\lat[\alpha,\beta,\gamma,\delta].
    \end{equation*}
\item
Addition of lattices is commutative and associative.
\item
Multiplication of lattices distributes over addition.
\item
If $\mLambda$ and $\mMu$ belong to $\ord[]$, then
$\ord[]\included\ord[\mLambda+\mMu]$. 
\item
If $\mLambda$ and $\mMu$ belong to $\roi$, then
$\ord[\mLambda+\mMu]=\roi$. 
  \end{enumerate}
  \end{xca}

  \begin{xca}
Show that $\lat[n,1+\omega]$ and $\lat[1,n\omega]$ both belong to
$\lat[1,n\omega]$ .
  \end{xca}

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\end{document}