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# Homework 8

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\Large\bf Homework 7 for Math 581F\\
Due Friday, November 30, 2007
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Each problem has equal weight, and parts of problems are worth the
same amount as each other.

\begin{enumerate}

\item Give a very detailed outline of your final project.
{\bf Your final project is due December 7, 2007.}

\item Let $K=\Q(\zeta_5)$ and let $r$ be the number of real embeddings
and $s$ the number of pairs of complex conjugate embeddings.
\begin{enumerate}
\item Show that $r=0$ and $s=2$.
\item Find explicit generators for the group of
units $U_K$.
\item Draw an illustration of the log map
$\vphi:U_K\to \R^2$, including the hyperplane
$x_1+x_2=0$ and the lattice in the hyperplane
spanned by the image of $U_K$.
\end{enumerate}

\item Let $n=6$.  For a number field $K$, let $e,f,g$ be the
ramification, residue class degree, and number of primes over $p$
for a rational prime $p$.

\begin{enumerate}
\item  Give an example of a number field $K$ of degree $6$ and
a prime $p$ such that $e=6$, or prove no such field exists.
\item  Give an example of a number field $K$ of degree $6$ and
a prime $p$ such that $f=6$, or prove no such field exists.
\item  Give an example of a number field $K$ of degree $6$ and
a prime $p$ such that $g=6$, or prove no such field exists.
\item  Give an example of a number field $K$ of degree $6$ and
a prime $p$ such that $e=f=2$, or prove no such field exists.
\end{enumerate}

\item
\begin{enumerate}
\item Give an example of a finite nontrivial Galois extension $K$ of $\Q$
and a prime ideal $\p$ such that $D_\p = \Gal(K/\Q)$.
\item Give an example of a finite nontrivial Galois extension $K$ of
$\Q$ and a prime ideal $\p$ such that $D_\p$ has order~$1$.
\item Give an example of a finite Galois extension~$K$ of
$\Q$ and a prime ideal $\p$ such that $D_\p$ is not a normal
subgroup of $\Gal(K/\Q)$.
\item Give an example of a finite Galois extension~$K$ of
$\Q$ and a prime ideal $\p$ such that $I_\p$ is not a normal
subgroup of $\Gal(K/\Q)$.
\end{enumerate}

\end{enumerate}

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2013-05-11 18:33