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


\begin{center}
  \textbf{Math 105}\\ Homework 2
\end{center}


\begin{enumerate}

\item (3-1-7) Universal Mapping Property for completions Let $L$ be a
  field complete with respect to a valuation $|\cdot |_L$ and $K$ a
  field with valuation $|\cdot |_K$, and $\widehat K$ its completion
  under the valuation.  Assume that there is an isometry
  $\lambda: K \to L$.  Show there is a unique isometric field
  embedding $\rho: \widehat K \to L$ making the diagram below commute.

  \begin{equation}
    \begin{tikzcd}[column sep=large, row sep=huge]
      \widetilde K \arrow[r, "\rho"] &L\\
      K \arrow[u, hook] \arrow[ur,"\lambda", swap]
    \end{tikzcd}
  \end{equation}

\item (3-1-18) Let $K$ be a field with valuation $|\cdot |_K$.  Show
  that $|\cdot |_K$ is non-archimedean iff there exists $C>0$ with
  $|m|_K\le C$ for all $m \in \Z$.


\item We showed in class that a number field $K$ embeds as a discrete
  subring of its adele ring, $\A_K$.
  \begin{enumerate}
  \item Show that $\A_K$ has a Hausdorff topology, so that it is a
    locally compact topological ring with a Hausdorff topology.
  \item Show the $K$ embeds as a closed subset of $\A_K$.
  \end{enumerate}

\item Let $G$ be a topological group, i.e., $G$ has a topology with
  respect to which the group operations are continuous.
  \begin{enumerate}
  \item Show that every open subgroup of $G$ is closed, and every
    closed subgroup of $G$ of finite index is open.
  \item Show that every subgroup $H\le G$ which contains a
    neighborhood of the identity is open.

  \item Show that if $H$ is s a subgroup of $G$, the coset space $G/H$
    is discrete if and only if $H$ is open.
  \end{enumerate}
\end{enumerate}

\end{document}


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