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\title{Math 129: Algebraic Number Theory\\
Homework Assignment 6}
\author{William Stein}
\date{Due: Thursday, March 25, 2004}
\begin{document}
\maketitle
\begin{enumerate}

\item Let $p\in \Z$ and let $K$ be a number field.  Show that 
$\Norm_{K/\Q}(p\O_K) = p^{[K:\Q]}$.


\item
A totally real number field is a number field in which all embeddings
into $\C$ have image in $\R$.  Prove there are totally real number
fields of degree $p$, for every prime $p$.  [Hint: Let $\zeta_n$
denote a primitive $n$th root of unity.  For $n\geq 3$, show that
$\Q(\zeta_n+1/\zeta_n)$ is totally real of degree $\vphi(n)/2$.  Now
prove that $\vphi(n)/2$ can be made divisible by any prime.]

\item Give an example of a number field $K/\Q$ and a prime $p$ such
that the $e_i$ in the factorization of $p\O_K$ are not all the same.

\item Let $K$ be a number field.  Give the ``simplest''
proof you can think of that there are only finitely many
primes that ramify (i.e., have some $e_i>1$) in $K$.
[The meaning of ``simplest'' is a matter of taste.]

\item Give examples to show that for $K/\Q$ a Galois extension, the
quantity $e$ can be arbirarily large and $f$ can be arbitrarily large.

\item Suppose $K/\Q$ is Galois and $p$ is a prime such that 
$p\O_K$ is also prime (i.e., $p$ is inert in $K$).  Show
that $\Gal(K/\Q)$ is a cyclic group.

\item (Problem 7, page 116, from Marcus {\em Number Fields})  For
each of the following, find
a prime $p$ and quadratic extensions $K$ and $L$ of $\Q$
that illustrates the assertion:
\begin{enumerate}
\item The prime $p$ can be totally ramified in $K$ and $L$ without
being totally ramified in $KL$.
\item The fields $K$ and $L$ can each contain unique primes lying over $p$
while $KL$ does not.
\item The prime $p$ can be inert in $K$ and $L$ without being inert in $KL$.
\item The residue field extensions of $\F_p$ can be trivial
for $K$ and $L$ without being trivial for $KL$.
\end{enumerate}

\end{enumerate}


\end{document}
