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\pagestyle{fancy} \lhead{{\sf Homework 14 $|$ Due April 26 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120 $|$ Visual Algebra $|$
    Spring 2024 $|$ 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 All of the isomorphism theorems for groups have analogues for
    rings. The proofs just amount to showing that the group
    homomorphisms are additionally ring homomorphisms, and we will
    carry out these details in this problem.
    \begin{enumerate}
      \item The \emph{fundamental homomorphism theorem} (FHT) says that
        $R/\Ker(\phi)\cong\Image(\phi)$. To prove this, we construct a
        map
        \[
        \iota\colon R/I\longto\Image(\phi),\qquad\quad\iota(r+I)=\phi(r),
        \]
        which we already know is a well-defined group
        isomorphism. Show that it is also a ring homomorphism.
      \item By the \emph{correspondence theorem}, every subgroup of
        $R/I$ has the form $J/I$ for some $I\leq J\leq R$. Show that
        $J/I$ is an ideal of $R/I$ if and only if $J$ is an ideal of
        $R$.

      \item The \emph{fraction theorem} says that $(R/I)/(J/I)\cong R/J$. This is proven by constructing a map
        \[
        \phi\colon R/I\longto R/J,\qquad\quad \phi(r+I)=r+J
        \]
        which we already know is a group homomorphism
        with $\Ker(\phi)=J/I$. Show that it is also a ring
        homomorphism, and then apply the FHT. 
      \item The \emph{diamond theorem} says that $(S+I)/I\cong
        S/(S\cap I)$ for a subring $S$ and ideal $I$.
        \begin{enumerate}[(i)]
        \item Prove that $S\cap I$ is an ideal of $S$.
        \item We already know that
          \[
          \phi\colon S\longto (S+I)/I,\qquad\quad \phi(s)=s+I
          \]
          is a group homomorphism with $\Ker(\phi)=S\cap I$. Show that
          it is also a ring homomorphism, and then apply the FHT. \medskip
        \end{enumerate}
          
    \end{enumerate}

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

\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
  $1/2\not\in A$. \medskip

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

  
\item Let $R$ be a commutative ring with $1$.
  \begin{enumerate}
    \item Show that if $x$ is contained in every maximal ideal, then
      $1+x$ is a unit.
    \item A ring is \emph{local} if it has a unique maximal
      ideal. Show that $R$ is local if and only if the non-units form
      an ideal.
    \item Let $p$ be a fixed prime. Show that the ring 
      \[
      R=\big\{\tfrac{a}{b}\mid a,b\in\Z,\;(a,b)=1,\; p\nmid b\big\}\subseteq\Q
      \]
      is local, and characterize units and maximal unique ideal.
  \end{enumerate}
  
  %%----------------------------------------------------------------

\end{enumerate}

\end{document}
