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\begin{center}
\textbf{Math 204B (Number Theory), UCSD, winter 2021 \\
Problem Set 17 -- due Thursday, March 11, 2021}
\end{center}

If you are planning to submit a final project, these will be due by Friday, March 19.

\begin{enumerate}
\item
Let $L/K$ be a Galois extension of number fields with Galois group $G$.
\begin{enumerate}
\item[(a)]
Give an explicit example for which $\Cl(K) \to \Cl(L)^G$ fails to be injective. (Hint: use the principal ideal theorem.)
\item[(b)]
Give an explicit example for which $\Cl(K) \to \Cl(L)^G$ fails to be surjective. (Hint: take $L$ to be a quadratic extension of $\QQ$ with $\Cl(L) \cong \ZZ/4\ZZ$.)
\end{enumerate}

\item
Put $K = \QQ(\sqrt{13}, \sqrt{17})$. Show that $5^2 \in \Norm_{K_w/\QQ_v} K_w^*$ for every place $w$ of $K$ lying above a place $v$ of $\QQ$, but $5^2 \notin \Norm_{K/\QQ} K^*$.

\item
Prove the following special case of the \emph{Hasse-Minkowski theorem}. Let $K$ be a number field and choose $a,b,c \in K^*$. Suppose that for each place $v$ of $K$, there exist
$x,y,z \in K_v$, not all zero, such that $ax^2 + by^2 + cz^2 = 0$. Then there exist $x,y,z \in K$, not all zero, such that $ax^2 + by^2 + cz^2$. 
(Hint: show that $-c$ is a norm from $K (\sqrt{-b/a})$ to $K$.) 

\item
\begin{enumerate}
\item[(a)]
Let $K$ be a field of characteristic not equal to 2.
Show that 16 is an 8th power in $K$ if and only if one of $-1,2, -2$ is a square in $K$.
\item[(b)]
Put $K = \QQ(\sqrt{7})$. 
Show that 16 is an 8th power in every completion of $K$, but not in $K$ itself.
\end{enumerate}

\item
Let $K$ be a number field, let $S$ be a finite set of finite places of $K$, and let $m$ be a positive integer. Prove that there exists a cyclic extension $L$ of $K$ contained in $K(\zeta_n)$ for some $n$ such that for all $v \in S$, 
for some place $w$ of $L$ above $K$, $[L_w:K_v]$ is divisible by $m$.

\end{enumerate}

\end{document}

