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\lhead{\texttt{MATH 204A}}
\chead{}
\rhead{\texttt{Fall 2024}}
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\title{Homework 2}
\author{due Friday, October 18, 2024 at 11:59pm}   
\date{}

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The homework is due in Gradescope at the stated time. You may submit handwritten text (scanned from paper or written on a tablet) or a typed PDF (e.g., created using LaTeX). If you choose the handwritten option, you have to also submit a hardcopy in class or put it in my mailbox. \\

\noindent Collaboration and research is fine, as long as you do the following.
\begin{itemize}
\item Try the problems yourself first!
\item Write the solutions in your own words.
\item Acknowledge all sources and collaborators.\\
\end{itemize}

\hrule

\vspace*{1cm}

\noindent For the Chinese remainder theorem for Dedekind domains, see for example Neukirch Theorem I.3.6.\\


\begin{enumerate}

\item
Let $R$ be a Dedekind domain. Prove that for every nonzero ideal $I$ of $R$, $R/I$ is a principal ideal ring. 
(Hint: use the Chinese remainder theorem to reduce to the case where $I$ is a power of a prime $\mathfrak{p}$.
In that case, choose $\pi \in \mathfrak{p} \setminus \mathfrak{p}^2$ and consider the powers of $\pi$.)\\

\item
Let $R$ be a Dedekind domain. Prove that every ideal of $R$ can be generated by at most two elements.\\

\item
Let $R$ be the ring $\ZZ[\alpha]/(\alpha^3 - \alpha - 1)$.
Show that the prime factorization of the principal ideal $23R$ is given by
\[
23R = (23, \alpha-10)^2 (23, \alpha-3).
\]
(In particular, you should show that the factors are indeed prime.)\\

\item
Prove that the constant $2^n$ in Minkowski's lattice point theorem cannot be improved.\\

\item
Let $K$ be a number field. Using the finiteness of the class group of $K$, prove that there exists a finite
extension $L$ of $K$ such that every ideal of $\mathcal{O}_K$ generates a principal ideal of $\mathcal{O}_L$.\\

\newpage

\vspace*{0.1cm}

\item
Let $p>2$ be a prime number and $K = \QQ(\zeta_p)$. 
\begin{enumerate}
\item
Compute $\Trace_{K/\QQ}(\zeta_p^j)$ for $j=0,\dots,p-1$.
\item
Compute $\Norm_{K/\QQ}(1-\zeta_p)$.
\item
Show that $(1-\zeta_p)\calO_K \cap \ZZ = p\ZZ$. (Note: we will use this later to show that $\calO_K = \ZZ[\zeta_p]$,
so don't assume this here.)
\end{enumerate}

\end{enumerate}

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