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\title{Elementary Number Theory II, examination III}
%\author{David Pierce}
\date{May 26, 2008}
\usepackage{verbatim}  % allows a comment environment:
%\begin{comment}
\address{Mathematics Dept\\
Middle East Technical University\\
Ankara 06531, Turkey}

\email{dpierce@metu.edu.tr}
\urladdr{http://www.math.metu.edu.tr/~dpierce/}
%\end{comment}

\usepackage{amsmath,amsthm,amssymb}
\usepackage{url}
\usepackage{textcomp}  % supposedly useful with \oldstylenums
\usepackage{hfoldsty} % this didn't work until I added missing
		      % brackets to some of the files.

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

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

%%%%%%%%%%%%%%%
\newcommand{\included}{\subseteq}      % [the name suggests the meaning here]

\newcommand{\stnd}[1]{\mathbb{#1}}
\newcommand{\Z}{\stnd{Z}}         % integers
\newcommand{\R}{\stnd{R}}         % reals
\newcommand{\Q}{\stnd{Q}}         % rationals

\newcommand{\mi}{\mathrm i}
%\newcommand{\gr}{\upphi}  % golden ratio
\newcommand{\mLambda}{\mathit{\Lambda}}
\newcommand{\mPi}{\mathit{\Pi}}
\newcommand{\mMu}{M}

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

\newcommand{\size}[1]{\lvert#1\rvert}


\let\oldsqrt\sqrt
\renewcommand{\sqrt}[2][1]{\oldsqrt{\vphantom{#1}}#2}
%\newcommand{\rft}{\sqrt{14}}
%\newcommand{\rtt}{\sqrt{13}}

\newcommand{\lat}[1][\alpha,\beta]{\langle#1\rangle} % lattice
\newcommand{\ord}[1][\mLambda]{\mathfrak O_{#1}}
\newcommand{\roi}{\ord[K]}

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\newtheorem{problem}{Problem}

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\newtheorem{solution}{Solution}

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%{\begin{proof}[Solution.]}%
%{\end{proof}}

%\numberwithin{equation}{section}
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\begin{document}
  \maketitle\thispagestyle{empty}

Solve these four problems in 90 minutes.  \emph{\.Iyi
  \c cal\i\c smalar.}  My solutions can be found at
\begin{center}
\url{http://www.metu.edu.tr/~dpierce/solu.pdf}\\
\url{http://www.metu.edu.tr/~dpierce/solu.ps}   
\end{center}


\vfill
\begin{problem}
  Suppose $\sqrt 2=[a_0;a_1,a_2,\dots]$, and as usual let
  $p_n/q_n=[a_0;a_1,\dots,a_n]$.  Find rational integers $a$, $b$,
  $k$, and $\ell$ such that
  \begin{equation*}
    p_n+q_n\sqrt 2=(a+b\sqrt 2)(k+\ell\sqrt 2)^n
  \end{equation*}
for all positive rational integers $n$.
\end{problem}
\vfill
\begin{problem}
Here $\mLambda$ and $\mMu$ are lattices in some quadratic field.
\begin{enumerate}
\item
  Find $\size{\mLambda/\mMu}$, that is, $(\mLambda:\mMu)$, when
  \begin{enumerate}
    \item
$\mLambda=\lat$, $\mMu=\lat[2\alpha,3\beta]$;
\item
$\mLambda=\lat$, $\mMu=\lat[2\alpha,\alpha+3\beta]$.
  \end{enumerate}
\item
Assuming $\mMu\included\mLambda$, find a number $n$ such that
$n\mLambda\included\mMu$. 
\end{enumerate}
\end{problem}
\vfill
\begin{problem}
  In some quadratic field, find a lattice $\mLambda$ such that
  $\norm{\mLambda}=1$, but $\mLambda\neq\ord$.
\end{problem}
\vfill
\begin{problem}
  Letting $K=\Q(\sqrt5)$ and $\ord[]=\roi$, for each $p$ in
  $\{2,3,5,7,11\}$, find the prime factorization of $p\ord[]$ in $\ord[]$.
\end{problem}
\vfill
\end{document}