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\pagestyle{fancy} \lhead{{\sf Homework 12 $|$ Due November 21 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2022 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}

  
\item Construct the Cayley tables, Cayley graphs, and subring lattices
  of the finite field $\F_9\cong\Z_3[x]/(x^2+x+2)$. Examples for the
  finite fields
  \[
  \F_4\cong\Z_2[x]/(x^2+x+1)\qquad\text{and}\qquad
  \F_8\cong\Z_2[x]/(x^3+x+1)
  \]
  are shown below. \vspace{-4mm}
  
  %% Finite field F_8, multiplication
  \[
  \tikzstyle{e-pp} = [draw, ultra thick, tpink]
  \begin{tikzpicture}[scale=1]
    %%
    \colorlet{color-000}{tpink}
    \colorlet{color-100}{tyellow}
    \colorlet{color-010}{tred}
    \colorlet{color-110}{tblue}
    \colorlet{color-001}{tpurple}
    \colorlet{color-101}{torange}
    \colorlet{color-011}{tgreen}
    \colorlet{color-111}{tlime}
    %%
    \colorlet{color-0}{tred}
    \colorlet{color-1}{tyellow}
    \colorlet{color-x}{tred}
    \colorlet{color-x+1}{tblue}
    %%
    \begin{scope}[shift={(,0)}]
      \path[fill=color-1] (-.1,3) rectangle ++(1,1);
      \path[fill=color-x] (-.1,2) rectangle ++(1,1);
      \path[fill=color-x+1] (-.1,1) rectangle ++(1,1);
      %%
      \path[fill=color-1] (1,4.1) rectangle ++(1,.9);
      \path[fill=color-x] (2,4.1) rectangle ++(1,.9);
      \path[fill=color-x+1] (3,4.1) rectangle ++(1,.9);
      %%
      \path[fill=color-x+1] (2,2) rectangle ++(1,1);
      \path[fill=color-x+1] (3,3) rectangle ++(1,1);
      %%
      \path[fill=color-1] (1,3) rectangle ++(1,1);
      \path[fill=color-x] (3,1) rectangle ++(1,1);
      %%
      \path[fill=color-x] (1,2) rectangle ++(1,1);
      \path[fill=color-1] (2,1) rectangle ++(1,1);
      %%
      \path[fill=color-x+1] (1,1) rectangle ++(1,1);
      \path[fill=color-x] (2,3) rectangle ++(1,1);
      \path[fill=color-1] (3,2) rectangle ++(1,1);
      %%
      \foreach \i in {1,2,3,4} {
        \draw [very thin] (\i,1) -- (\i,\n-1); 
        \draw [very thin] (\i,\n-1+.1) -- (\i,\n); 
        \draw [very thin] (1,\i) -- (\n-1,\i); 
        \draw [very thin] (-.1,\i) -- (.9,\i); 
      } 
      \draw [very thin] (1,\n-1+.1) rectangle (\n-1,\n);
      \draw [very thin] (-.1,1) rectangle (.9,\n-1); 
      \node at (0.4,4.6) {\Large $\times$};
      \node at (0.4,3.5) {$1$};
      \node at (0.4,2.5) {$x$};
      \node at (0.4,1.5) {$x\!+\!1$};
      \node at (1.5,4.5) {$1$}; 
      \node at (2.5,4.5) {$x$};
      \node at (3.5,4.5) {$x\!+\!1$};
      %%
      \node at (1.5,3.5) {$1$};
      \node at (1.5,2.5) {$x$};
      \node at (1.5,1.5) {$x\!+\!1$};
      %%
      \node at (2.5,3.5) {$x$};
      \node at (2.5,2.5) {$x\!+\!1$};
      \node at (2.5,1.5) {$1$};
      %%
      \node at (3.5,3.5) {$x\!+\!1$};
      \node at (3.5,2.5) {$1$};
      \node at (3.5,1.5) {$x$};
    \end{scope}
    %%
    \begin{scope}[scale=1,shift={(6.5,0)}]
      \renewcommand*{\n}{8}%
      \tikzstyle{every node}=[font=\footnotesize]
      %% Left column of table (header)
      \path[fill=color-100] (-.1,7) rectangle ++(1,1);
      \path[fill=color-010] (-.1,6) rectangle ++(1,1);
      \path[fill=color-110] (-.1,5) rectangle ++(1,1);
      \path[fill=color-001] (-.1,4) rectangle ++(1,1);
      \path[fill=color-101] (-.1,3) rectangle ++(1,1);
      \path[fill=color-011] (-.1,2) rectangle ++(1,1);
      \path[fill=color-111] (-.1,1) rectangle ++(1,1);
      %% Top row of table (header)
      \path[fill=color-100] (1,8.1) rectangle ++(1,1);
      \path[fill=color-010] (2,8.1) rectangle ++(1,1);
      \path[fill=color-110] (3,8.1) rectangle ++(1,1);
      \path[fill=color-001] (4,8.1) rectangle ++(1,1);
      \path[fill=color-101] (5,8.1) rectangle ++(1,1);
      \path[fill=color-011] (6,8.1) rectangle ++(1,1);
      \path[fill=color-111] (7,8.1) rectangle ++(1,1);
      %
      \path[fill=color-100] (1,7) rectangle ++(1,1);
      \path[fill=color-010] (2,7) rectangle ++(1,1);
      \path[fill=color-110] (3,7) rectangle ++(1,1);
      \path[fill=color-001] (4,7) rectangle ++(1,1);
      \path[fill=color-101] (5,7) rectangle ++(1,1);
      \path[fill=color-011] (6,7) rectangle ++(1,1);
      \path[fill=color-111] (7,7) rectangle ++(1,1);
      %
      \path[fill=color-010] (1,6) rectangle ++(1,1);
      \path[fill=color-001] (2,6) rectangle ++(1,1);
      \path[fill=color-011] (3,6) rectangle ++(1,1);
      \path[fill=color-110] (4,6) rectangle ++(1,1);
      \path[fill=color-100] (5,6) rectangle ++(1,1);
      \path[fill=color-111] (6,6) rectangle ++(1,1);
      \path[fill=color-101] (7,6) rectangle ++(1,1);
      %
      \path[fill=color-110] (1,5) rectangle ++(1,1);
      \path[fill=color-011] (2,5) rectangle ++(1,1);
      \path[fill=color-101] (3,5) rectangle ++(1,1);
      \path[fill=color-111] (4,5) rectangle ++(1,1);
      \path[fill=color-001] (5,5) rectangle ++(1,1);
      \path[fill=color-100] (6,5) rectangle ++(1,1);
      \path[fill=color-010] (7,5) rectangle ++(1,1);
      %
      \path[fill=color-001] (1,4) rectangle ++(1,1);
      \path[fill=color-110] (2,4) rectangle ++(1,1);
      \path[fill=color-111] (3,4) rectangle ++(1,1);
      \path[fill=color-011] (4,4) rectangle ++(1,1);
      \path[fill=color-010] (5,4) rectangle ++(1,1);
      \path[fill=color-101] (6,4) rectangle ++(1,1);
      \path[fill=color-100] (7,4) rectangle ++(1,1);
      %
      \path[fill=color-101] (1,3) rectangle ++(1,1);
      \path[fill=color-100] (2,3) rectangle ++(1,1);
      \path[fill=color-001] (3,3) rectangle ++(1,1);
      \path[fill=color-010] (4,3) rectangle ++(1,1);
      \path[fill=color-111] (5,3) rectangle ++(1,1);
      \path[fill=color-110] (6,3) rectangle ++(1,1);
      \path[fill=color-011] (7,3) rectangle ++(1,1);
      %
      \path[fill=color-011] (1,2) rectangle ++(1,1);
      \path[fill=color-111] (2,2) rectangle ++(1,1);
      \path[fill=color-100] (3,2) rectangle ++(1,1);
      \path[fill=color-101] (4,2) rectangle ++(1,1);
      \path[fill=color-110] (5,2) rectangle ++(1,1);
      \path[fill=color-010] (6,2) rectangle ++(1,1);
      \path[fill=color-001] (7,2) rectangle ++(1,1);
      %
      \path[fill=color-111] (1,1) rectangle ++(1,1);
      \path[fill=color-101] (2,1) rectangle ++(1,1);
      \path[fill=color-010] (3,1) rectangle ++(1,1);
      \path[fill=color-100] (4,1) rectangle ++(1,1);
      \path[fill=color-011] (5,1) rectangle ++(1,1);
      \path[fill=color-001] (6,1) rectangle ++(1,1);
      \path[fill=color-110] (7,1) rectangle ++(1,1);
      %%
      \foreach \i in {1,...,\n} {
        \draw [very thin] (\i,1) -- (\i,\n); 
        \draw [very thin] (\i,\n+.1) -- (\i,\n+1.1); 
        \draw [very thin] (1,\i) -- (\n,\i); 
        \draw [very thin] (-.1,\i) -- (.9,\i); 
      }
      \draw [very thin] (1,\n+.1) rectangle (\n,\n+1.1);
      \draw [very thin] (-.1,1) rectangle (.9,\n);
      %%
      \node at (0.4,8.6) {\Large $\times$};
      \node at (0.4,7.5) {$1$};
      \node at (0.4,6.5) {$x$};
      \node at (0.4,5.5) {$x\!+\!1$}; 
      \node at (0.4,4.5) {$x^2$}; 
      \node at (0.4,3.5) {$x^2\!+\!1$};
      \node at (0.4,2.5) {$x^2\!+\!x$};
      \node at (0.4,1.5) {\tiny $x^2\hspace{-1mm}+\hspace{-1mm}x\hspace{-1mm}+\hspace{-1mm}1$};
      %% 
      \node at (1.5,8.6) {$1$};
      \node at (2.5,8.6) {$x$};
      \node at (3.5,8.6) {$x\!+\!1$}; 
      \node at (4.5,8.6) {$x^2$}; 
      \node at (5.5,8.6) {$x^2\!+\!1$};
      \node at (6.5,8.6) {$x^2\!+\!x$};
      \node at (7.5,8.6) {\tiny $x^2\hspace{-1mm}+\hspace{-1mm}x\hspace{-1mm}+\hspace{-1mm}1$};
      %%%%%%
      \node at (1.5,7.5) {$1$};
      \node at (1.5,6.5) {$x$};
      \node at (1.5,5.5) {$x\!+\!1$}; 
      \node at (1.5,4.5) {$x^2$}; 
      \node at (1.5,3.5) {$x^2\!+\!1$};
      \node at (1.5,2.5) {$x^2\!+\!x$};
      \node at (1.5,1.5) {\tiny $x^2\hspace{-1mm}+\hspace{-1mm}x\hspace{-1mm}+\hspace{-1mm}1$};
      %%
      \node at (2.5,7.5) {$x$};
      \node at (2.5,6.5) {$x^2$};
      \node at (2.5,5.5) {$x^2\!+\!x$};
      \node at (2.5,4.5) {$x\!+\!1$}; 
      \node at (2.5,3.5) {$1$}; 
      \node at (2.5,2.5) {\tiny $x^2\hspace{-1mm}+\hspace{-1mm}x\hspace{-1mm}+\hspace{-1mm}1$};
      \node at (2.5,1.5) {$x^2\!+\!1$};
      %%
      \node at (3.5,7.5) {$x\!+\!1$};
      \node at (3.5,6.5) {$x^2\!+\!x$};
      \node at (3.5,5.5) {$x^2\!+\!1$};
      \node at (3.5,4.5) {\tiny $x^2\hspace{-1mm}+\hspace{-1mm}x\hspace{-1mm}+\hspace{-1mm}1$};
      \node at (3.5,3.5) {$x^2$};
      \node at (3.5,2.5) {$1$};
      \node at (3.5,1.5) {$x$};
      %%
      \node at (4.5,7.5) {$x^2$};
      \node at (4.5,6.5) {$x\!+\!1$};
      \node at (4.5,5.5) {\tiny $x^2\hspace{-1mm}+\hspace{-1mm}x\hspace{-1mm}+\hspace{-1mm}1$}; 
      \node at (4.5,4.5) {$x^2\!+\!x$};
      \node at (4.5,3.5) {$x$};
      \node at (4.5,2.5) {$x^2\!+\!1$};
      \node at (4.5,1.5) {$1$};
      %%
      \node at (5.5,7.5) {$x^2\!+\!1$};
      \node at (5.5,6.5) {$1$};
      \node at (5.5,5.5) {$x^2$};
      \node at (5.5,4.5) {$x$}; 
      \node at (5.5,3.5) {\tiny $x^2\hspace{-1mm}+\hspace{-1mm}x\hspace{-1mm}+\hspace{-1mm}1$};
      \node at (5.5,2.5)  {$x\!+\!1$};
      \node at (5.5,1.5) {$x^2\!+\!x$};
      %%
      \node at (6.5,7.5) {$x^2\!+\!x$};
      \node at (6.5,6.5) {\tiny $x^2\hspace{-1mm}+\hspace{-1mm}x\hspace{-1mm}+\hspace{-1mm}1$};
      \node at (6.5,5.5) {$1$}; 
      \node at (6.5,4.5) {$x^2\!+\!1$}; 
      \node at (6.5,3.5) {$x\!+\!1$};
      \node at (6.5,2.5) {$x$};
      \node at (6.5,1.5) {$x^2$};
      %%
      \node at (7.5,7.5) {\tiny $x^2\hspace{-1mm}+\hspace{-1mm}x\hspace{-1mm}+\hspace{-1mm}1$};
      \node at (7.5,6.5) {$x^2\!+\!1$};
      \node at (7.5,5.5) {$x$}; 
      \node at (7.5,4.5) {$1$}; 
      \node at (7.5,3.5) {$x^2\!+\!x$};
      \node at (7.5,2.5) {$x^2$};
      \node at (7.5,1.5) {$x+1$};
      %%
    \end{scope}
    %%
  \end{tikzpicture}
  \]
  
  \vspace{-5mm}
  
  \[
  \colorlet{Ideal}{Blue}
  \colorlet{Subring}{Red}
  \colorlet{Neither}{faded}
  \begin{tikzpicture}[shorten >= -3pt, shorten <= -3pt, scale=2]
    \begin{scope}[shift={(0,0)}]
      \tikzstyle{every node}=[font=\normalsize]
      \node[Ideal] (G) at (0,2) {$\F_4$};
      \node[Neither] (a) at (-1,1) {$\Z_2$};
      \node[Subring] (b) at (0,1) {$\F_2$};
      \node[Neither] (c) at (1,1) {$\Z_2$};
      \node[Ideal] (0) at (0,0) {$\<0\>$};
      \draw (G) to (b); \draw[faded] (G) to (a); \draw[faded] (G) to (c);
      \draw (0) to (b); \draw[faded] (0) to (a); \draw[faded] (0) to (c);
    \end{scope}
    %%
    \begin{scope}[shift={(4,0)},xscale=.8]
      \tikzstyle{every node}=[font=\normalsize]
      \node[Ideal] (G) at (0,3) {\normalsize $\F_8$};
      %%
      \node[Neither] (010-001) at (-3,2) {$\Z_2^2$}; % 010,001,011
      \node[Neither] (100-001) at (-2,2) {$\Z_2^2$}; % 100,001,101    
      \node[Neither] (100-010) at (-1,2) {$\Z_2^2$}; % 100,010,110
      \node[Neither] (100-011) at (0,2) {$\Z_2^2$};  % 100,011,111
      \node[Neither] (010-101) at (1,2) {$\Z_2^2$}; % 010,101,111
      \node[Neither] (110-001) at (2,2) {$\Z_2^2$};  % 110,001,111
      \node[Neither] (110-011) at (3,2) {$\Z_2^2$};  % 110,011,101
      %%
      \node[Neither] (100) at (-3,1) {$\Z_2$};
      \node[Neither] (010) at (-2,1) {$\Z_2$};
      \node[Neither] (001) at (-1,1) {$\Z_2$};
      \node[Subring] (011) at (0,1) {$\F_2$};
      \node[Neither] (101) at (1,1) {$\Z_2$};
      \node[Neither] (110) at (2,1) {$\Z_2$};
      \node[Neither] (111) at (3,1) {$\Z_2$};
      %%
      \node[Ideal] (000) at (0,0) {$\<0\>$};
      %%
      \draw[faded] (000) to (100); \draw[faded] (000) to (010); \draw[faded] (000) to (110);
      \draw[faded] (000) to (101); \draw[faded] (000) to (001); \draw (000) to (011);
      \draw[faded] (000) to (111);
      %%
      \draw[faded](100)to(100-010); \draw[faded](100)to(100-001); \draw[faded](100)to(100-011);
      \draw[faded](010)to(100-010); \draw[faded](010)to(010-001); \draw[faded](010)to(010-101);
      \draw[faded](110)to(100-010); \draw[faded] (110)to(110-001); \draw [faded] (110)to(110-011);
      \draw[faded] (101)to(100-001); \draw[faded](101)to(010-101); \draw[faded] (101)to(110-011);
      \draw[faded](111)to(100-011); \draw[faded](111)to(010-101); \draw[faded] (111)to(110-001);
      \draw[faded](011)to(010-001); \draw(011)to(100-011); \draw[faded] (011)to(110-011);
      \draw[faded](001)to(100-001); \draw[faded](001)to(010-001); \draw[faded] (001)to(110-001);
      %%
      \draw[faded] (100-010) to (G); \draw[faded] (100-001) to (G); \draw[faded] (010-001) to (G);
      \draw (100-011) to (G); \draw[faded] (010-101) to (G); \draw[faded] (110-001) to (G);
      \draw[faded] (110-011) to (G);
    \end{scope}
  \end{tikzpicture}
  \]
  
  \vspace{-4mm}
  
  
  \tikzstyle{g-dashed} = [draw,very thick, darkgreen, dashed, -stealth]
  \tikzstyle{p} = [draw, very thick, dashed, amethyst,-stealth]
  \[
  \begin{tikzpicture}[scale=1]
    \begin{scope}[shift={(0,0)},scale=1.2]
      \tikzstyle{v-big} = [circle, draw, fill=lightgrey,inner sep=0pt, 
        minimum size=7mm]
      \tikzstyle{every node}=[font=\footnotesize]
      \node (0) at (135:1.5) [v-big] {$0$};
      \node (1) at (45:1.5) [v-big] {$1$};
      \node (3) at (-135:1.5) [v-big] {$x$};
      \node (2) at (-45:1.5) [v-big] {\scriptsize $x\!+\!1$};   
      \draw [bb] (0) to (1);
      \draw [bb] (2) to (3);
      \draw [rr] (0) to (3);
      \draw [rr] (1) to (2);
      \draw [g-dashed,bend right=20] (1) to (3);
      \draw [g-dashed,bend right=40] (2) to (1);
      \draw [g-dashed,bend right=40] (3) to (2);
    \end{scope}
    %%
    \begin{scope}[shift={(5,-1.2)},scale=1.2]
      \tikzstyle{every node}=[font=\scriptsize]
      \tikzstyle{v-big} = [circle, draw, fill=lightgrey,inner sep=0pt, 
        minimum size=7mm]
      \node (2) at (0,0) [v-big] {$x^2\!\!+\!1$};
      \node (0) at (-0.1,2) [v-big] {$1$};
      \node (3) at (2,-.4) [v-big] {\tiny \scalebox{.85}{$x^2\!\!+\!\!x\!\!+\!\!1$}};
      \node (1) at (2.1,1.7) [v-big] {$x\!+\!1$};
      \node (6) at (1,1.1) [v-big] {$x^2$};
      \node (4) at (.9,3) [v-big] {$0$};
      \node (7) at (2.7,.8) [v-big] {$x^2\!\!+\!x$};
      \node (5) at (2.8,2.8) [v-big] {$x$}; 
      \draw [rr] (0) to (1);
      \draw [rr] (2) to (3);
      \draw [bb] (0) to (4);
      \draw [bb] (1) to (5);
      \draw [bb] (2) to (6);
      \draw [bb] (3) to (7);
      \draw [gg] (0) to (2);
      \draw [gg] (1) to (3);
      \draw [rr] (4) to (5);
      \draw [rr] (6) to (7);
      \draw [gg] (4) to (6);
      \draw [gg] (5) to (7);
      \draw [p,bend left=10] (0) to (5);
      \draw [p,bend right=15] (5) to (6);
      \draw [p,bend left=0] (6) to (1);
      \draw [p,bend left=0] (1) to (7);
      \draw [p,bend left=25] (7) to (3);
      \draw [p,bend left=20] (3) to (2);
      \draw [p,bend left=20] (2) to (0);
    \end{scope}
  \end{tikzpicture}
  \]
  
  %%----------------------------------------------------------------
  
\item Use Zorn's lemma to show that the ring $\R$ contains a subring
  $A$ containing $1$ that is maximal with respect to the property that
  $1/2\not\in A$. \\
  
  %%----------------------------------------------------------------
    
\item Let $R$ be a commutative ring with $1$. The \emph{radical} of an ideal 
  $I\subseteq R$ is the set
  \[
  \sqrt{I}=\big\{x\in R\mid x^n\in I\text{ for some } n\in\N\big\},
  \]
  and $I$ is a \emph{radical ideal} if $I=\sqrt{I}$. An ideal
  $I\subsetneq R$ is \emph{primary} if $ab\in I$ implies $a\in I$ or
  $b^n\in I$ for some $n\in\N$.
  \begin{enumerate}
  \item Show that an ideal $I$ is prime if and only if it is primary
    and radical.
  \item Characterize the radical of a primary ideal.
  \item Suppose that $R$ is a PID. Characterize its nonzero primary
    ideals.
  \item Show that $I$ is primary if and only if all zero-divisors in
    $R/I$ are nilpotent.
  \item Give an example to show that, in general, there are primary
    ideals which are not prime powers. \\
  \end{enumerate}

  %%----------------------------------------------------------------

\item Let $R$ be commutative ring. For a multiplictive semigroup
  $D\subseteq R$ containing no zero divisors, let $\mathcal{F}=R\times
  D$ and define a relation on $\mathcal{F}$ where $(a,b)\sim(c,d)$ if
  $ad=bc$.
  \begin{enumerate}
  \item Show that $\sim$ is an equivalence relation on $\mathcal{F}$.
  \item Denote the equivalence class of $(a,b)$ by $a/b$ and the set
    of equivalence classes by $D^{-1}R$ (called the \emph{localization} of
    $R$ at $D$). Show that $D^{-1}R$ is a commutative ring with $1$.
  \item If $a\in S$, show that $\{ra/a\mid r\in R\}$ is a subring of
    $D^{-1}R$ and that $r\mapsto ra/a$ is a monomorphism, therefy
    identifying $R$ with a subring of $D^{-1}R$.
  \item Under this identification, show that every $d\in D$ is a unit
    in $D^{-1}R$.
  \item State and prove a ``co-universal'' definition for $D^{-1}R$. \\
  \end{enumerate}

  %%----------------------------------------------------------------
  
\item Let $R$ be an integral domain and $P\subseteq R$ a prime ideal.
  \begin{enumerate} 
  \item Show that both $P$ and $R\setminus P$ are multiplicative
    semigroups.  
  \item Let $R_P:=D^{-1}R$ be the ring of fractions for $D=R\setminus
    P$. Show that $U(R_P)=R_P\setminus R_PP$. Conclude that $R_PP$ is
    the unique maximal ideal in $R_D$.
  \item Describe this construction for the prime ideal
    $(p)\subseteq\Z$, and then for $(x)\subseteq\Z[x]$.
  \end{enumerate}


  
  
\end{enumerate}

\end{document}




%% Later HW (PIDs, prime elements, polynomials)
\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 $\Z[x]$.
\end{enumerate}
