# Math 581f Midterm

The midterm as a pdf: midterm.pdf

Here's the raw source, in case you are latexing your solutions:

\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 Take Home Midterm for Math 581F\\ Due Monday November 5, 2007, at the start of class. \end{center} There are five problems. Do not talk to other people about the problems, though you may email me if you have questions about problems (in case they are ill-stated). You may use Sage if you very explicitly describe exactly what calculations you did, and you may consult any book or web page (cite it). Make sure to do a good job on this exam, since it is {\em worth 30\% of your grade}, almost the same as your entire homework. And it's not a terribly hard exam either. I fully expect everbody to get a perfect score. \begin{enumerate} \item \begin{enumerate} \item Consider the map $\vphi: \Z^3 \to \Z^4$ given by $$\vphi((1,0,0)) = (1,2,3,0), \quad \vphi((0,1,0)) = (4,5,6,0),\quad \vphi((0,0,1)) = (7,8,9,0).$$ Write the cokernel of $\vphi$, i.e., $\Z^4/\Im(\vphi)$ as a direct sum of cyclic groups. \item Which of the following rings are Noetherian? \begin{enumerate} \item $\Z[\pi, e, \sqrt{2}]$ -- adjoin some real numbers to $\Z$, \item $\F_7(x_1,x_2,x_3)[y_1, y_2]$ -- adjoin some variables to a finite field, \item $\Z[x_1, x_2, \ldots, x_n, \ldots, n\geq 1]/(x_1-x_2)$ -- adjoin infinitely many variables to $\Z$ and quotient out by the ideal generated by $x_1 - x_2$, \item $\overline{\Z}[x]$ -- adjoin a variable to the ring of all algebraic integers in a fixed choice of algebraic closure of $\QQ$. \end{enumerate} \end{enumerate} \item \begin{enumerate} \item Prove that if $K$ is a number field then there are infinitely many prime ideals of $\O_K$. \item Suppose $I$ and $J$ are fractional ideals of a number field $K$. \begin{enumerate} \item Prove that $I+J$ is a fractional ideal. \item Prove that if $I=\prod \mathfrak{p}_i^{e_i}$ and $J= \prod \mathfrak{p}_i^{f_i}$, then $I +J = \prod_{i=1}^r \mathfrak{p}_i^{\min(e_i, f_i)}$. \end{enumerate} \end{enumerate} \item Let $\O_K$ be the ring of integers of the number field $K=\Q(\sqrt{3^{997} - 1})$. [Note: You will probably not get anywhere trying to create this number field directly in the current version Sage.] \begin{enumerate} \item Find an order $R$ such that $[\O_K:R]$ is not divisible by $7$. \item Give generators for the prime ideal factors of the ideal $7\O_K$. \end{enumerate} \item Let $K=\Q(\sqrt{-23})$, and set $a=\sqrt{-23}\in K$.. Show how to find $n \in \ZZ$ and $\alpha \in \O_K$, such that $$ \left( 2a + 2, a - 11, -\frac{5}{2}a - \frac{17}{2}\right) \O_K = (n, \alpha)\O_K. $$ Explain the steps you use to find $n$ and $\alpha$ -- don't just ``ask Sage''. \item Make a list of every integral ideal $I$ of the ring of integers of $K=\Q(\sqrt{-23})$ that has norm at most $10$. \end{enumerate} \end{document}