\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym}
\usepackage[all]{xy}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{patterns}
\usetikzlibrary{shapes.geometric}
\usepackage{tkz-graph}
\usepackage{tikz-cd}
\usepackage{enumerate}
\usepackage{multicol}
\usepackage{url}
\usepackage{bm} 

\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in


% Useful math macros
\def\ds{\displaystyle}
\def\bold{\bm}     % Change ``bm'' to ''mathbf'' for upright boldface.

\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}
\newcommand{\vvvv}[4]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \end{bmatrix}}
\newcommand{\vvvvv}[5]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \\ #5 \end{bmatrix}}
\def\<{\langle}
\def\>{\rangle}
\newcommand{\ceil}[1] {\left\lceil #1 \right\rceil}
\newcommand{\floor}[1] {\left\lfloor #1 \right\rfloor}
\def\normal{\lhd}
\def\normaleq{\unlhd}
\def\nnormal{\ntriangleleft}
\def\nnormaleq{\ntrianglelefteq}
\def\Dot{\bm{.}}

%\def\subgroup{\leqslant}
%\def\nsubgroup{\nleqslant}
%\def\normal{\leqslant}
%\def\nnormal{\nleqslant}

\newcommand{\rddots}{\rotatebox{90}{$\ddots$}}
\def\setmin{-}

\def\Sum{\sum\limits}
\def\Prod{\prod\limits}
\def\Cap{\mathop\cap\limits}
\def\Cup{\mathop\cup\limits}
\def\Times{\mathop\times\limits}
\def\Oplus{\mathop\oplus\limits}
\def\Lim{\mathop\lim\limits}

\def\And{\wedge}
\def\Or{\vee}
\def\Not{\neg}

\def\longto{\longrightarrow}
\def\into{\hookrightarrow}
\def\longinto{\longhookrightarrow}
\def\onto{\twoheadrightarrow}


\def\A{\mathbb{A}}
\def\C{\mathbb{C}}
\def\F{\mathbb{F}}
\def\HH{\mathbb{H}}
\def\N{\mathbb{N}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\Z{\mathbb{Z}}


\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\Lcm}{lcm}
\DeclareMathOperator{\del}{\nabla}
\DeclareMathOperator{\Trace}{tr}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\Image}{Im}
\DeclareMathOperator\Perm{Perm}
\DeclareMathOperator\Char{char}
\DeclareMathOperator\Aut{Aut}
\DeclareMathOperator\Inn{Inn}
\DeclareMathOperator\Out{Out}
\DeclareMathOperator\orb{orb}
\DeclareMathOperator\Orb{Orb}
\DeclareMathOperator\Stab{Stab}
\DeclareMathOperator\stab{stab}
\DeclareMathOperator\Fix{Fix}
\DeclareMathOperator\fix{fix}
\DeclareMathOperator\Syl{Syl}
\DeclareMathOperator\Pre{Pre}
\DeclareMathOperator\Ker{Ker}
\DeclareMathOperator\Sym{Sym}
\DeclareMathOperator\Mat{Mat}
\DeclareMathOperator{\cl}{cl}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator\PGL{PGL}
\DeclareMathOperator\PSL{PSL}
\DeclareMathOperator\PSU{PSU}
\DeclareMathOperator\PSP{PsP}
\DeclareMathOperator\Heis{Heis}
\DeclareMathOperator\Pauli{Pauli}
\DeclareMathOperator\Dih{D}
\DeclareMathOperator\Dic{Dic}
\DeclareMathOperator\DQ{DQ}
\DeclareMathOperator\SD{SD}
\DeclareMathOperator\SA{SA}
\DeclareMathOperator\AGL{AGL}
\DeclareMathOperator\Gal{Gal}
\DeclareMathOperator\Deg{Deg}

%% Categories
\DeclareMathOperator\Hom{Hom}
\DeclareMathOperator\Ob{Ob}
\newcommand{\catname}[1]{{\normalfont\textbf{#1}}}
\newcommand{\Ab}{\catname{Ab}}
\newcommand{\Field}{\catname{Field}}
\newcommand{\Set}{\catname{Set}}
\newcommand{\Grp}{\catname{Grp}}
\newcommand{\Semi}{\catname{Sgp}}
\newcommand{\Ring}{\catname{Ring}}
\newcommand{\Rng}{\catname{Rng}}
\newcommand{\Top}{\catname{Top}}
\newcommand{\Vect}{\catname{Vect}}
\newcommand{\Nil}{\catname{Nil}}
\newcommand{\Svl}{\catname{Svl}}
\newcommand\cat{\mathcal{C}}
\newcommand\dat{\mathcal{D}}
\newcommand\eat{\mathcal{E}}
\newcommand\fun{\mathcal{F}}

\DeclareMathOperator\Rect{\mathbf{Rect}}
\DeclareMathOperator\Tri{\mathbf{Tri}}
\DeclareMathOperator\Coin{\mathbf{Coin}}
\DeclareMathOperator\BOX{\mathbf{Box}}
\DeclareMathOperator\Light{\mathbf{Light}}
\DeclareMathOperator\Sq{\mathbf{Sq}}

\DeclareMathOperator\Frieze{\mathbf{Frz}}

% Colors 
\usepackage{xcolor}
\usepackage{color}
\definecolor{darkblue}{rgb}{0, 0, .6}
\definecolor{grey}{rgb}{.7, .7, .7}
\newcommand{\balert}[1]{\textcolor{blue}{#1}}
\newcommand{\galert}[1]{\textcolor{darkgreen}{#1}}
\newcommand{\Palert}[1]{\textcolor{amethyst}{#1}}
\newcommand{\Alert}[1]{\textcolor{Red}{#1}}
\newcommand{\Balert}[1]{\textcolor{Blue}{#1}}
\newcommand{\Galert}[1]{\textcolor{darkgreen}{#1}}


%\hypersetup{colorlinks=true,linkcolor=darkblue,anchorcolor=darkblue,
%  citecolor=darkblue,pagecolor=darkblue,urlcolor=darkblue,
%  pdftitle={},pdfauthor={}
%}                               %% THIS CAUSES COMPILER ERROR
\colorlet{lightred}{red!30!white}
\colorlet{lightblue}{blue!30!white}
\colorlet{lightyellow}{yellow!30!white}
\colorlet{keylime}{green!10!white}
\colorlet{darkgreen}{green!50!black}
\colorlet{midgreen}{green!70!black}
\colorlet{lightgreen}{darkgreen!30!white}
\colorlet{lightpurple}{violet!60!white}
\colorlet{darkgrey}{black!70}
\colorlet{midgrey}{black!50}
\definecolor{Red}{rgb}{.9,0,0}
\definecolor{Blue}{rgb}{0,0,.9}
\colorlet{Green}{green!50!black}
\definecolor{Gold}{rgb}{.83,.67,.08}

\definecolor{conj-purple}{HTML}{E4B7FF}
\definecolor{conj-orange}{HTML}{FFD4B7}

\colorlet{lred}{red!50!white}
\definecolor{lblue}{rgb}{0.60, 0.75, 1}
\colorlet{lgreen}{green!40!white}
\colorlet{lyellow}{yellow!30!white}
%\definecolor{lpurple}{rgb}{0.6 0.35 0.8}
\definecolor{lpurple}{rgb}{0.7 0.45 0.9}
\colorlet{lorange}{orange!50!white}
\definecolor{lightgrey}{rgb}{.85, .85, .85}
\definecolor{faded}{rgb}{.75,.75,.75}
\definecolor{midgrey}{rgb}{.5,.5,.5}
%\definecolor{midgrey}{rgb}{.35, .35, .35}
%\definecolor{amethyst}{rgb}{0.6 0.4 0.8}
\definecolor{amethyst}{rgb}{0.54 0.17 0.89}
\definecolor{amber}{rgb}{1.0 0.49 0.0}
\definecolor{turq}{rgb}{0.12 0.69 0.71}

\colorlet{lightestgray}{gray!15!white}

\colorlet{lightestblue}{blue!15!white}
\colorlet{midblue}{blue!45!white}

\colorlet{lightestred}{red!15!white}
\colorlet{midred}{red!45!white}
\colorlet{lightestgreen}{darkgreen!15!white}
\colorlet{lightestpurple}{amethyst!20!white}



% COLORS FOR MULTIPLICATION TABLES  
\colorlet{tred}{red!70!white}
\colorlet{tblue}{blue!60!white}     %(higher number is more faded)
\colorlet{tgreen}{green!75!black}   %(higher number is brighter)
\colorlet{tyellow}{yellow!60!white}
\colorlet{tlime}{green!80!white}
\colorlet{tpink}{pink}
\colorlet{tpurple}{lpurple}
\colorlet{torange}{orange}

\newcommand*{\n}{5}   % counter for Cayley tables

%% CAYLEY DIAGRAM GLOBAL DEFAULTS
\tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=6mm]
\tikzstyle{r} = [draw, very thick, Red, -stealth]        % Red --->
\tikzstyle{b} = [draw, very thick, Blue, -stealth]       % Blue --->
\tikzstyle{g} = [draw, very thick, darkgreen, -stealth]  % Green --->
\tikzstyle{p} = [draw, very thick, amethyst, -stealth]     % Purple --->
\tikzstyle{rr} = [draw, very thick, Red]                 % Red ----
\tikzstyle{bb} = [draw, very thick, Blue]                % Blue ----
\tikzstyle{gg} = [draw, very thick, darkgreen]           % Green ----
\tikzstyle{oo} = [draw, very thick, orange]           % Purple ----
\tikzstyle{pp} = [draw, very thick, amethyst]           % Purple ----
\tikzstyle{tt} = [draw, very thick, turq]           % Turquoise ----
\tikzstyle{yy} = [draw, very thick, Gold]           % Gold ----

\tikzstyle{R} = [draw, very thick, Red, -stealth']        % Red --->
\tikzstyle{B} = [draw, very thick, Blue, -stealth']       % Blue --->
\tikzstyle{G} = [draw, very thick, darkgreen, -stealth']  % Green --->

\tikzset{r/.style={draw, very thick, Red, -stealth}}        % Red --->
\tikzset{R/.style={draw, very thick, Red, -stealth'}}        % Red --->

\tikzset{v/.style={circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=6mm}}

\renewcommand\labelitemi{\rule[0.12em]{0.4em}{0.4em}}
\renewcommand\labelitemii{\normalfont\bfseries \rule[0.15em]{0.3em}{0.3em}}
\renewcommand\labelitemiii{\textasteriskcentered}
\renewcommand\labelitemiv{\textperiodcentered}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 14 $|$ Due December 9 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2022 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}


\item 
  \begin{enumerate}
  \item Solve the congruences
    \[
    x\equiv 1 \pmod{8}\,,\qquad
    x\equiv 3 \pmod{7}\,,\qquad
    x\equiv 9 \pmod{11}\,
    \]
    simultaneously for $x$ in the ring $\Z$ of integers.
  \item Solve the congruences
    \[
    x\equiv i \pmod{i+1}\,,\qquad
    x\equiv 1 \pmod{2-i}\,,\qquad
    x\equiv 1+i \pmod{3+4i}\,
    \]
    simultaneously for $x$ in the ring $R_{-1}$ of Gaussian integers. 
  \item Solve the congruences
    \[
    f(x)\equiv 1 \pmod{x-1}\,,\qquad
    f(x)\equiv x \pmod{x^2+1}\,,\qquad
    f(x)\equiv x^3 \pmod{x+1}\,
    \]
    simultaneously for $f(x)$ in $F[x]$, where $F$ is a field in which
    $1+1\neq 0$. \medskip
  \end{enumerate}


  
%% 
\item Let $R$ be a commutative ring with identity and $\F$ a field.
  \begin{enumerate}
  \item Show that if $R$ is a PID then any nonzero prime ideal
    $P\subseteq R$ is a maximal.
  \item Show that there is a bijective correspondence between maximal
    ideals of $\F[x]$ and monic irreducible polynomials in $\F[x]$.
  \item Show that if $M\subsetneq \Z[x]$ is a maximal ideal, then
    $M\bigcap\Z=(p)$ for some prime $p\neq 0$.
  \item Show that there is a bijective correspondence between maximal
    ideals of $\Z[x]$ that contain $p$ and monic irreducible polynomials
    in $\Z_p[x]$.
  \item Characterize all maximal ideals of $\Z[x]$. \medskip
\end{enumerate}
  

%
\item Consider the following rings $R_i$, for $i=1,\dots,6$, which are
  additionally $\C$-vector spaces:

  \hspace{15mm} $R_1 = \C[x]/(x^3-1)$
    
  \hspace{15mm} $R_2 = \C \times \C \times \C$
    
  \hspace{15mm} $R_3 = $ the ring of upper triangular $2 \times 2$ matrices over
    $\C$
    
  \hspace{15mm} $R_4 = \C[x]/(x-1) \times \C[x]/(x+i)\times\C[x]/(x-i)$
    
  \hspace{15mm} $R_5 = \C[x]/(x^2+1) \times \C[x]/(x-1)$
    
  \hspace{15mm} $R_6 = \C[x]/(x+1)^2 \times \C[x]/(x-1)$.

    \begin{enumerate} 
    \item Compute the dimension of each $R_i$ as a $\C$-vector space by
      giving an explicit basis.
    \item Partition the rings $R_1,\dots,R_6$ into isomorphism classes. \medskip
    \end{enumerate}


  
  \item For a fixed $a\in R$, denote the polynomial \emph{evaluation map} by
    \[
    \phi_a\colon R[x]\longto R,\qquad\phi_a\colon f(x)\longmapsto f(a).
    \]
    \begin{enumerate}
    \item If $I=(3,x^2+1)$ in $\Z[x]$, show that $\phi_a(I)=\Z$.
    \item Show by example how Part~(c) can fail if $3$ is replaced with
      a different odd prime $p$.   
    \item Given a polynomal, $f(x)\in R[x]$, substituting $a\in R$ for
      $x$ determines a \emph{polynomial function} $f\colon R\to R$,
      where $a\mapsto f(a)$. Show that if $R$ is an infinite integral
      domain, then the mapping $f(x)\mapsto f$ assiging to each
      polynomial in $R[x]$ to its corresponding polynomial function is
      1--1.
    \end{enumerate}
    
  
  
\end{enumerate}

\end{document}






