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\title{Math 129: Algebraic Number Theory\\
Homework Assignment 7}
\author{William Stein}
\date{Due: Thursday, April 8, 2004}
\begin{document}
\maketitle
\begin{enumerate}
\item Let $S_3$ by the symmetric group on three symbols, which
has order $6$.
\begin{enumerate}
\item \label{ex:a} Observe that $S_3\isom D_3$, where $D_3$ is the dihedral group
of order $6$, which is the group of symmetries of an equilateral
triangle.
\item Use (\ref{ex:a}) to write down an explicit
embedding $S_3\hra \GL_2(\C)$.
\item Let $K$ be the number field $\Q(\sqrt[3]{2},\omega)$,
where $\omega^3=1$ is a nontrivial cube root of unity.  Show
that $K$ is a Galois extension with Galois group isomorphic to $S_3$.
\item We thus obtain a $2$-dimensional irreducible complex
Galois representation
$$
\rho:\Gal(\Qbar/\Q) \to \Gal(K/\Q)\isom S_3 \subset \GL_2(\C).
$$
Compute a representative matrix of $\Frob_p$ and the characteristic polynomial
of $\Frob_p$ for $p=5,7,11,13$.  
\end{enumerate}

\item Let $K=\Q(\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7})$. 
Show that $K$ is Galois over $\Q$, compute the Galois
group of $K$, and compute $\Frob_{37}$.

\item Decide on a final project for this course.  Here are
some possible ideas, though you need not do a project on
one of these.  Joint projects are a possibility (see me).
\begin{enumerate}
\item How to compute class groups of number fields.
\item How to compute the unit group of a number field (we
didn't even prove the unit group is computable in class).
\item How to solve the norm equation $\Norm_{K/\Q}(x)=d$.
\item Explore relations between quadratic reciprocity and class
field theory for $\Q$.
\item The Chebotarev Density Theorem: read about it and explain
what the point is, and something about why it is true (e.g., for 
quadratic fields).
\item Give a proof of Dirichlet's theorem on primes in an arithmetic 
progression (connected to the Chebotarev project above).
\item Connection between ideal class groups of quadratic
imaginary fields and classes of positive definite binary
quadratic forms.  Gauss's class number problem.
\item The conjecture that there are infinitely many number fields
of class number 1.  What is known?  What do the Cohen-Lenstra
heuristics predict?  Why is this problem so hard?
\item Quadratic imaginary fields and complex multiplication
elliptic curves.
\item Elements of Shafarevich-Tate groups: give complete examples with
proofs of equations like $3x^3+4y^3+5z^3=0$ that have a solution (with
not all $x,y,z$ zero) over every $p$-adic field $\Q_p$ and over $\R$,
but not over $\Q$.
\end{enumerate}

\end{enumerate}


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