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\begin{flushright}
  Your Name Here
\end{flushright}


\begin{center}
  \textbf{Math 105}\\ Homework 4
\end{center}

\begin{enumerate}

\item (3-3-12) Let $K$ be a number field, and $\O_K^\times$ its unit
  group. Show that $\varepsilon \in \O_K^\times$ iff
  $N_{K/\Q}(\varepsilon) = \pm 1$.


\item (3-3-12) Let $D\in \Z$, squarefee $D < 0$, and set
  $K = \Q(\sqrt D)$. Let $i = \sqrt{-1}$ and $\omega$ a primitive cube
  root of unity in $\C$.  Show that the units of $K$ form a finite
  cyclic group.  In particular, show
  \[\O_K^\times =
    \begin{cases}
      \la i\ra =\{\pm 1, \pm i\}&\mbox{ if } D=-1,\\
      \la -\omega\ra= \{\pm 1, \pm\omega, \pm \omega^2\} &\mbox{ if } D=-3,\\
      <-1>=\{\pm 1\}&\mbox{ otherwise}.
    \end{cases}
  \]
 
\item (3-3-12) Let $D\in \Z$, squarefee $D > 1$, and set
  $K = \Q(\sqrt D)$.
  \begin{enumerate}
  \item Show that
    $\O_K^\times \cong \{\pm 1\} \times \la \varepsilon\ra$ where
    $\varepsilon$ has infinite order.
  \item The unique generator $\varepsilon$ which satisfies
    $\varepsilon >1$ is called the fundamental unit of $K$.  Show that
    the fundamental unit of $K = \Q(\sqrt 2)$ is
    $\varepsilon=1+\sqrt 2$.  Note: it is not always so easy: The
    fundamental unit of $\Q(\sqrt{94})$ is
    $\varepsilon=2143295+221064\sqrt{94}$.

  \end{enumerate}
\end{enumerate}

\end{document}


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