# Homework number 4 - Due Friday, October 26, 2007

\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 4 for Math 581F\\ Due FRIDAY October 26, 2007 \end{center} Each problem has equal weight, and parts of problems are worth the same amount as each other. \begin{enumerate} \item \begin{enumerate} \item Find by hand and with proof the ring of integers of each of the following two fields: $\Q(\sqrt{5})$, $\Q(i)$. \item Find the ring of integers of $\Q(x^5+7x+1)$ using a computer. \end{enumerate} \item Let $\O_K$ be the ring of integers of a number field $K$, and let $p\in\Z$ be a prime number. What is the cardinality of $\O_K/(p)$ in terms of $p$ and $[K:\Q]$, where $(p)$ is the ideal of $\O_K$ generated by~$p$? \item Explicitly factor the ideals generated by each of $2$, $3$, and $13$ in the ring of integers of $\Q(\sqrt[3]{2})$. (Thus you'll factor three separate integral ideals as products of prime ideals.) You may assume that the ring of integers of $\Q(\sqrt[3]{2})$ is $\Z[\sqrt[3]{2}]$, but do {\em not} simply use a computer command to do the factorizations. \item Let $K=\Q(\zeta_{13})$,where $\zeta_{13}$ is a primitive $13$th root of unity. Note that~$K$ has ring of integers $\O_K=\Z[\zeta_{13}]$. \begin{enumerate} \item Factor $2$, $3$, $5$, $7$, $11$, and $13$ in the ring of integers $\O_K$. You may use a computer. \item For $p\neq 13$, find a conjectural relationship between the number of prime ideal factors of $p\O_K$ and the order of the reduction of~$p$ in $(\Z/13\Z)^*$. \item Compute the minimal polynomial $f(x)\in\Z[x]$ of $\zeta_{13}$. Reinterpret your conjecture as a conjecture that relates the degrees of the irreducible factors of $f(x)\pmod{p}$ to the order of $p$ modulo~$13$. Does your conjecture remind you of quadratic reciprocity? \end{enumerate} \item Let $p$ be a prime. Let $\O_K$ be the ring of integers of a number field~$K$, and suppose $a\in \O_K$ is such that $[\O_K:\Z[a]]$ is finite and coprime to~$p$. Let $f(x)$ be the minimal polynomial of~$a$. We proved in class that if the reduction $\overline{f}\in\F_p[x]$ of $f$ factors as $$ \overline{f} = \prod g_i^{e_i}, $$ where the $g_i$ are distinct irreducible polynomials in $\F_p[x]$, then the primes appearing in the factorization of $p\O_K$ are the ideals $(p,g_i(a))$. In class, we did not prove that the exponents of these primes in the factorization of $p\O_K$ are the $e_i$. Prove this. \item \begin{enumerate} \item Give an example of a cubic {\em Galois} extension $K$ of $\QQ$. Use Sage to factor each prime $p<100$ (or more) in $\O_K$ and record the number of prime factors of each $p$. \item Give an example of a cubic {\em non-Galois} extension $K$ of $\QQ$. Use Sage to factor each prime $p<100$ (or more) in $\O_K$ and record the number of prime factors of each $p$. \item Come up with a more refined conjecture about the proportion of primes $p$ for which the number of prime factors is $1$, $2$ or $3$ in each of the above two cases. \end{enumerate} \end{enumerate} \end{document}