# Homework 8

\documentclass{article} \include{macros} \voffset=-0.05\textheight \textheight=1.1\textheight \hoffset=-0.05\textwidth \textwidth=1.1\textwidth \begin{document} \begin{center} \Large\bf Homework 7 for Math 581F\\ Due Friday, November 30, 2007 \end{center} 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} \end{document}