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\begin{document}

\begin{center}
\textbf{Math 204B (Number Theory), UCSD, winter 2021 \\
Problem Set 13 -- due Thursday, February 11, 2021}
\end{center}

\begin{enumerate}

\item
Let $K$ be a finite extension of $\QQ_p$.
\begin{enumerate}
\item[(a)]
Show that $K^*$ is a locally compact abelian group for the subspace topology on $K \times K$ via the map $x \mapsto (x,x^{-1})$.
\item[(b)]
Show that $K^*$ is closed in $K \times K$ but not in $K$. This distinction will show up more seriously in the definition of the id\`eles.
\end{enumerate}

\item
Let $L/K$ be an extension of finite fields. Prove directly (without using any group cohomology) that the norm map
$\Norm_{L/K}: L^* \to K^*$ is surjective. (Hint: write an explicit formula for the norm map.)

\item
Let $G$ be a finite group. Let $M$ be a $G$-module whose underlying abelian group is a $\QQ$-vector space. Prove that $M$ is acyclic.
(Hint: first use the description using cochains to show that the groups $H^i(G,M)$ are themselves divisible. Then combine with 
the fact that the cohomology groups are killed by $\#G$; see CFT Example 3.2.21.)

\item
Let $K$ be a finite extension of $\QQ_p$. 
\begin{enumerate}
\item[(a)]
Show that for any positive integer $n$, $K^*/(K^*)^n$ is a finite group.
\item[(b)]
Let $L$ be a finite extension of $K$.
Show that $\Norm_{L/K} L^*$ is an open subgroup of $K^*$ of finite index. (Hint: restrict the norm map to $K^*$, then apply (a).)
\end{enumerate}

\item
Let $p$ be an odd prime. 
Let $R_1$ and $R_2$ be quaternion algebras over $\QQ_p$ (see CFT exercise 4.1.5). Show directly (without local class field theory)
that if neither $R_1$ nor $R_2$ is isomorphic to
$M_2(\QQ_p)$, then they are isomorphic to each other. (Hint: first recall the structure of $\QQ_p^*/(\QQ_p^*)^2$.) Optional: extend this to $p=2$.

\end{enumerate}

\end{document}

