# ANT 07 Homework 2

\documentclass{article} \include{macros} \begin{document} \begin{center} \Large\bf Homework 2 for Math 581F, Due FRIDAY October 12, 2007\end{center} Each problem has equal weight, and parts of problems are worth the same amount as each other. \begin{enumerate} \item Let $\vphi:R\to S$ be a homomorphism of (commutative) rings. \begin{enumerate} \item Prove that if $I\subset S$ is an ideal, then $\vphi^{-1}(I)$ is an ideal of~$R$. \item Prove moreover that if $I$ is prime, then $\vphi^{-1}(I)$ is also prime. \end{enumerate} \item Let $\O_K$ be the ring of integers of a number field. The Zariski topology on the set $X=\Spec(\O_K)$ of all prime ideals of $\O_K$ has closed sets the sets of the form $$ V(I) = \{ \p\in X : \p \mid I\}, $$ where~$I$ varies through {\em all} ideals of $\O_K$, and $\p\mid I$ means that $I \subset \p$. \begin{enumerate} \item Prove that the collection of closed sets of the form $V(I)$ is a topology on $X$. \item Prove that the conclusion of (a) is still true if $\O_K$ is replaced by an order in $\O_K$, i.e., a subring that has finite index in $\O_K$ as a $\Z$-module. \end{enumerate} \item Let $\alpha = \sqrt{2} + \frac{1+\sqrt{5}}{2}$. \begin{enumerate} \item Is $\alpha$ an algebraic integer? \item Explicitly write down the minimal polynomial of $\alpha$ as an element of $\QQ[x]$. \end{enumerate} \item Which are the following rings are orders in the given number field. \begin{enumerate} \item The ring $R = \ZZ[i]$ in the number field $\QQ(i)$. \item The ring $R = \ZZ[i/2]$ in the number field $\QQ(i)$. \item The ring $R = \ZZ[17i]$ in the number field $\QQ(i)$. \item The ring $R = \ZZ[i]$ in the number field $\QQ(\sqrt[4]{-1})$. \end{enumerate} \item Give an example of each of the following, with proof: \begin{enumerate} \item A non-principal ideal in a ring. \item A module that is not finitely generated. \item The ring of integers of a number field of degree~$3$. \item An order in the ring of integers of a number field of degree~$5$. \item A non-diagonal matrix of left multiplication by an element of~$K$, where~$K$ is a degree~$3$ number field. \item An integral domain that is not integrally closed in its field of fractions. \item A Dedekind domain with finite cardinality. \item A fractional ideal of the ring of integers of a number field that is not an integral ideal. \end{enumerate} \end{enumerate} \end{document}