%%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[all]{xy} \usepackage{amsxtra} \usepackage[pdftex, breaklinks, linktocpage=true, bookmarksopen=true,bookmarksopenlevel=0,bookmarksnumbered=true]{hyperref} \hypersetup{ pdftitle = {Math 371 Assignment 1}, 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}{8.5in} %%% \setlength{\textwidth}{6.5in} %%% \setlength{\headsep}{0in} %%% \setlength{\headheight}{0in} %%% %\setlength{\footskip}{0in} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\End}{End} \newcommand{\Q}{\mathbf{Q}} \newcommand{\R}{\mathbf{R}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\C}{\mathbf{C}} \newcommand{\F}{\mathbf{F}} \newcommand{\p}{\mathfrak{p}} \begin{document} \begin{center} {\bf \Large \underline{Honors Algebra 4, MATH 371 Winter 2010}}\\ {\large Assignment 4}\\Due Wednesday, February 17 at 08:35 \end{center} \begin{enumerate} \item Let $R$ be a commutative ring with $1\neq 0$. \begin{enumerate} \item Prove that the nilradical of $R$ is equal to the intersection of the prime ideals of $R$. Hint: it's easy to show using the definition of prime that the nilradical is contained in every prime ideal. Conversely, suppose that $f$ is not nilpotent and consider the set $S$ of ideals $I$ of $R$ with the property that ``$n>0\implies f^n\not\in I$." Show that $S$ has maximal elements and that any such maximal element must be a prime ideal. \item Suppose that $R$ is {\em reduced}, {\em i.e.} that the nilradical of $R$ is the zero ideal. If $\p$ is a minimal prime ideal of $R$, show that the localization $R_{\p}$ has a unique prime ideal and conclude that $R_{\p}$ is a field. \item Again supposing $R$ to be reduced, prove that $R$ is isomorphic to a subring of a direct product of fields. \end{enumerate} \item Let $R$ be a commutative ring with $1\neq 0$ and let $\varphi:R\rightarrow R$ be a ring homomorphism. If $R$ is noetherian and $\varphi$ is surjective, show that $\varphi$ must be injective too, and hence an isomorphism. (Hint: Consider the iterates of $\varphi$ and their kernels.) Can you give a counter-example to this when $R$ is not noetherian? \item As usual, for a prime $p$ we write $\F_p=\Z/p\Z$ for the field with $p$ elements. \begin{enumerate} \item Find all monic irreducible polynomials in $\F_p[X]$ of degree $\le 3$ for $p=2,3,5$.\label{calc} \item Prove that for $f\in \F_p[X]$ monic and irreducible, the ideal $(f(X))$ is maximal and hence that $\F_p[X]/(f(X))$ is a field. Show that $\F_p[X]/(f(X))$ has finite cardinality $p^{\deg{f}}$ and use part (\ref{calc}) to explicitly construct finite fields of orders $8,9,25,125$. \item Prove that $\F_{7}[X]/(X^2+2)$ and $\F_7[X]/(X^2+X+3)$ are both finite fields of size 49. Show that these fields are isomorphic by exhibiting an explicit isomorphism between them. \end{enumerate} \item Let $R$ ba a ring with $1\neq 0$ and $M$ an $R$-module. Show that if $N_1\subseteq N_2\subseteq \cdots $ is an ascending chain of submodules of $M$ then $\cup_{i\ge 1} N_i$ is a submodule of $N$. Show by way of counterexample that modules over a ring need not have maximal proper submodules (in contrast to the special case of ideals in a ring with $1$). \item Let $R$ be any commutative ring with $1\neq 0$ and $M$ and $R$-module. Show that the canonical map $$\Hom_R(R,M)\rightarrow M$$ sending $\varphi$ to $\varphi(1)$ is an isomorphism of $R$-modules. \item Let $F=\R$ and let $V=\R^3$. Consider the linear map $\varphi:V\rightarrow V$ given by rotation through an angle of $\pi/2$ about the $z$-axis. Consider $V$ as an $F[X]$-module by defining $$(a_nX^n+a_{n-1}X^{n-1}+\cdots + a_1X + a_0)v :=(a_n\varphi^n+a_{n-1}\varphi^{n-1}+\cdots + a_1\varphi + a_0)v,$$ where $\varphi^i$ is the composition of $\varphi$ with itself $i$-times. \begin{enumerate} \item What are the $F[X]$-submodules of $V$? \item Show that $V$ is naturally a module over the quotient ring $F[X]/(X^3-X^2+X-1)$. \end{enumerate} \item Let $R$ be a ring with $1\neq 0$. \begin{enumerate} \item For a left ideal $I$ of $R$ and an $R$-module $M$, define $$IM:=\left\{ r_1m_1 + r_2m_2 + \cdots + r_k m_k\ :\ r_i\in R,\ m_i\in M,\ k\in \Z_{\ge 0}\right\}.$$ Show that $IM$ is an $R$-submodule of $M$. \item Prove that for any ideal $I$ of $R$ and any positive integer $n$, there is a canonical isomorphism of $R$-modules $$R^n/IR^n \simeq R/IR \times R/IR \times \cdots \times R/IR$$ with $n$-factors in the product on the right.\label{fieldtrick} \item Suppose now that $R$ is commutative and that $R^n\simeq R^m$ as $R$-modules. Show that $m=n$. Hint: reduce to the case of finite dimensional vector spaces over a field by applying (\ref{fieldtrick}) with $I$ a maximal ideal of $R$. \item If $R$ is commutative and $A$ is any finite set of cardinality $n$, show that $F(A)\simeq R^n$ as $R$-modules (Hint: Show that $R^n$ satisfies the same universal mapping property as $F(A)$ and deduce from this that one has maps in both directions whose composition in either order must be the identity). Conclude that the rank of a free module over a commutative ring is well-definied if it is finite. \end{enumerate} \item Let $R$ be a ring with $1\neq 0$ and $M$ an $R$-module. We say that $M$ is {\em irreducible} if $M\neq 0$ and the only submodules of $M$ are $0$ and $M$. \begin{enumerate} \item Show that $M$ is irreducible if and only if $M$ is a nonzero cyclic $R$-module. \item If $R$ is commutative, show that $M$ is irreducible if and only if $M\simeq R/I$ as $R$-modules for some maximal ideal $I$ of $R$. \item Prove Schur's lemma: if $M_1$ and $M_2$ are irreducible $R$-modules then any nonzero $R$-module homomorphism $\varphi:M_1\rightarrow M_2$ is an isomorphism. \end{enumerate} \end{enumerate} \end{document}