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\title{Math 129: Algebraic Number Theory\\
Homework Assignment 3}
\author{William Stein}
\date{Due: Thursday, March 4, 2004}
\begin{document}
\maketitle

\noindent{\bf The problems:}
\begin{enumerate}


\item 
\begin{enumerate}
\item Suppose $K$ is a number field of degree $2$.  Prove that
$\O_K=\Z[a]$ for some $a\in\O_K$.
\item Prove that if $K$ and $K'$ are two number fields
of degree $2$ and $\Disc(\O_K) = \Disc(\O_{K'})$ then
$K=K'$.
\end{enumerate}

\item (*) Does there exist a number field~$K$ of degree~$4$ such that
$\O_K\neq \Z[a]$ for all $a\in\O_K$?  If so, give an explicit example.

\item Let $K$ be the quintic number field generated by a root of
$x^5+7x^4+3x^2-x+1$.  Draw a diagram (be creative) that illustrates
the factorization of every prime $p\in\Z$, with $p<100$, in $\O_K$.

\item (Problem 1.9 in Swinnerton-Dyer) Show that the only solutions
$x,y\in\Z$ to $y^2 = x^3 - 13$
are given by $x=17, y=\pm 70$, as follows. 
Factor the equation $y^2+13=x^3$ 
in the number field $\Q(\sqrt{-13})$, which has
class number~$2$.  Show that if $x,y$ is an integer solution then the
ideal $(y+\sqrt{-13})$ must be the cube of an ideal, and hence
$y+\sqrt{-13} = (a+b\sqrt{-13})^3$; thus $1=b(3a^2-13b^2)$.

%\item Find representative ideals for each element of the class group
%of $\Q(\sqrt{-23})$.

\item Suppose $I$ and $J$ are ideals in the ring $\O_K$ of integers of
a number field $K$.  Does $IJ=I\cap J$?  Prove or give a
counterexample.

\item Let $\O_K$ be the ring of integers $\Q(\sqrt{5})$,
and let 
$$I=(5, 2+\sqrt{5})\qquad\text{and}\qquad J=(209,(389+\sqrt{5})/2)$$
be integral ideals of $\O_K$.
\begin{enumerate}
\item 
Find an element of $\O_K$ that is congruent to $\sqrt{5}$
modulo $I$ and is congruent to $1-\sqrt{5}$ modulo $J$.
\item What is the cardinality of $(\O_K/I) \oplus (\O_K/J)$?
\item Find an element $a\in I$ such that $(a)/I$ is coprime
to $J$.
\end{enumerate}

\item Let $\O_K$ be the ring of integers of a number field
$K$, and suppose $K$ has exactly $2s$ complex embeddings. Prove
that the sign of $\Disc(\O_K)$ is $(-1)^s$.

%\item (*) Suppose $\O$ is an order in the ring of integers $\O_K$ of a
%number field.  Is every ideal in $\O$ necessarily generated by two
%elements?

\end{enumerate}


\end{document}
