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


\begin{center}
  \textbf{Math 105}\\ Homework 3
\end{center}

(3-2-2) (Understanding the topology on the ideles) Let $K$ be a
  number field.  We know that the adeles of $K$, $\A_K$, is a
  topological ring, as is each factor $K_\nu$ with respect to their
  respective topologies.  Establish the following:
  \begin{enumerate}
  \item For a non-archimedean place $\nu$ of $K$, show that $K_\nu$ is
    a topological field.  Here one need only show that for
    $\alpha\in K_\nu^\times$, $\alpha \mapsto \alpha^{-1}$ is continuous.
  \item Show that in the subspace topology on $\A_K^\times$, the map
    $\widetilde \alpha \mapsto \widetilde\alpha^{-1}$ is not
    continuous.
  \item If instead, we embed $\A_K^\times$ into $\A_K\times \A_K$ via
    $\widetilde \alpha \mapsto (\widetilde \alpha,
    \widetilde\alpha^{-1})$, then inversion is continuous with respect
    to the subspace topology inherited from $\A_K\times \A_K$.
  \end{enumerate}


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