%%This is a standard LaTeX2e article document template. personal version 12/5/200%% \documentclass[11pt,twoside]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%packages%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \pagestyle{empty} \usepackage{latexsym} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amstext} \usepackage{multicol} \usepackage{amsxtra} \usepackage[pdftex, breaklinks, linktocpage=true, bookmarksopen=true,bookmarksopenlevel=0,bookmarksnumbered=true]{hyperref} \hypersetup{ pdftitle = {Math 371 Assignment 3}, pdfauthor = {\textcopyright\ Bryden Cais}, pdfcreator = {\LaTeX\ with package \flqq hyperref\frqq}, colorlinks = {true} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%formatting%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\topmargin}{0in} %%% This sets all the spacing stuff to use the page more \setlength{\oddsidemargin}{0in} %%% efficiently than the normal "article" setup would. \setlength{\evensidemargin}{0in} %%% It's OK to play with these some. \setlength{\textheight}{7.5in} %%% \setlength{\textwidth}{6.5in} %%% \setlength{\headsep}{0in} %%% \setlength{\headheight}{0in} %%% %\setlength{\footskip}{0in} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Frac}{Frac} \newcommand{\Q}{\mathbf{Q}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\C}{\mathbf{C}} \newcommand{\F}{\mathbf{F}} \begin{document} \begin{center} {\bf \Large \underline{Honors Algebra 4, MATH 371 Winter 2010}}\\ {\large Assignment 3}\\{Due Friday, February 5 at 08:35} \end{center} \begin{enumerate} \item Let $R\neq 0$ be a commutative ring with 1 and let $S\subseteq R$ be the subset of nonzero elements which are not zero divisors. \begin{enumerate} \item Show that $S$ is multiplicatively closed. \item By definition, {\em the total ring of fractions of $R$} is the ring $\Frac(R):=S^{-1}R$; it is a ring equipped with a canonical ring homomorphism $R\rightarrow S^{-1}R$. If $T$ is any multiplicatively closed subset of $R$ that is contained in $S$, show that there is a canonical injective ring homomorphism $T^{-1}R\rightarrow \Frac(R)$, and conclude that $T^{-1}R$ is isomorphic to a subring of $\Frac(R)$. \item If $R$ is a domain, prove that $\Frac(R)$ is a field and hence that $T^{-1}R$ is a domain for any $T$ as above. \end{enumerate} \item Let $R$ be a commutative ring with $1$. \begin{enumerate} \item Let $S\subseteq R$ be a multiplicatively closed subset. Prove that the prime ideals of $S^{-1}R$ are in bijective correspondence with the prime ideals of $R$ whose intersection with $S$ is empty. \item If $\mathfrak{p}$ is an ideal of $R$, show that $S:=R\setminus \mathfrak{p}$ is a multiplicatively closed subset if and only if $\mathfrak{p}$ is a prime ideal. Writing $R_{\mathfrak{p}}$ for the ring of fractions $S^{-1}R$, show that $R_{\mathfrak{p}}$ has a unique maximal ideal, and that this ideal is the image of $\mathfrak{p}$ under the canonical ring homomorphism $R\rightarrow R_{\mathfrak{p}}$. (In other words, the {\em localization of $R$ at $\mathfrak{p}$} is a {\em local ring}). \item Let $r\in R$ be arbitrary. Show that the following are equivalent: \begin{enumerate} \item $r=0$ \item The image of $r$ in $R_{\mathfrak{p}}$ is zero for all prime ideals $\mathfrak{p}$ of $R$. \item The image of $r$ in $R_{\mathfrak{p}}$ is zero for all maximal ideals $\mathfrak{p}$ of $R$. \end{enumerate} \end{enumerate} \item Do exercises 8--11 in \S7.6 of Dummit and Foote (inductive and projective limits). \item A {\em B\'ezout domain} is an integral domain in which every finitely generated ideal is principal. \begin{enumerate} \item Show that a B\'ezout domain is a PID if and only if it is noetherian. \item Let $R$ be an integral domain. Prove that $R$ is a Bezout domain if and only if every pair of elements $a,b\in R$ has a GCD $d\in R$ that can be written as an $R$-linear combination of $a$ and $b$, {\em i.e.} such that there exist $x,y\in R$ with $d=ax+by$. \item Prove that a ring $R$ is a PID if and only if it is a B\'ezout domain that is also a UFD. \item Let $R$ be the quotient ring of the polynomial ring $\Q[x_0,x_1,\ldots]$ over $\Q$ in countably many variables by the ideal $I$ generated by the set $\{x_i-x_{i+1}^2\}_{i\ge 0}$. Show that $R$ is a B\'ezout domain which is not a PID (Hint: have a look at Dummit and Foote, \S9.2 \# 12). {\bf Remark:} The above example of a B\'ezout domain which is not a PID is somewhat artificial. More natural examples include the ``ring of algebraic integers" ({\em i.e.} the set of all roots of monic irreducible polynomials in one variable over $\Z$) and the ring of holomorphic functions on the complex plane. The proofs that these are B'ezout domains is, as far as I know, difficult. For example, in the case of the algebraic integers, one needs the theory of class groups). \end{enumerate} \item Let $R=\Z[i]:=\Z[X]/(X^2+1)$ be the ring of {\em Gaussian integers}. \begin{enumerate} \item Let $N:R\rightarrow \Z_{\ge 0}$ be the {\em field norm}, that is $$N(a+bi) := (a+bi)(a-bi)=a^2+b^2.$$ Prove that $R$ is a Euclidean domain with this norm. Hint: there is a proof in the book on pg. 272, but you should try to find a different proof by thinking {\em geometrically}. \item Show that $N$ is multiplicative, {\em i.e.} $N(xy)=N(x)N(y)$ and deduce that $u\in R$ is a unit if and only if $N(u)=1$. Conclude that $R^{\times}$ is a cyclic group of order 4, with generator $\pm i$. \item Let $p\in \Z$ be a (positive) prime number. If $p\equiv 3\bmod 4$, show that $p$ is prime in $\Z[i]$ and that $\Z[i]/(p)$ is a finite field of characteristic $p$ which, as a vector space over $\F_p$, has dimension 2. If $p=2$ or $p\equiv 1\bmod 4$, prove that $p$ is not prime in $\Z[i]$, but is the norm of a prime $\mathfrak{p}\in \Z[i]$ with $\Z[i]/(\mathfrak{p})$ isomorphic to the finite field $\F_p$. Conclude that $p\in \Z$ can be written as the sum of two integer squares if and only if $p=2$ or $p\equiv 1\bmod 4$. \end{enumerate} \end{enumerate} \end{document}