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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due February 3 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4130/6130 $|$ Visual Algebra $|$
    Spring 2023 $|$ M.~Macauley}}

\begin{document}

%$\;$ \vspace{-4mm}

\begin{enumerate}

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

\item A picture illustrating the quadratic integers
  $R_{-5}=\Z[\sqrt{-5}]=\big\{a+b\sqrt{-5}\mid a,b\in\Z\big\}$ as a
  subring of $\C$ is shown below, with the primes in black, and
  non-prime irreducibles colored.
  %% R_{-5}
\[
  \begin{tikzpicture}[scale=.6]
  \tikzstyle{p} = [circle, draw, fill=black,inner sep=0pt, 
    minimum size=1.5mm]
  \tikzstyle{i} = [circle, draw, red,fill=red,inner sep=0pt, 
    minimum size=1.5mm]
  \tikzstyle{c} = [circle, draw, Purple,fill=Purple,inner sep=0pt, 
    minimum size=1.5mm]    
  %%    
  \draw [lightestblue!40,fill=lightestblue!40] (0:0) -- (11.35,0) -- (11.35,11.35) -- (0,11.35) -- cycle;
  \draw [lightestred!30,fill=lightestred!30] (0:0) -- (-11.35,0) -- (-11.35,11.35) -- (0,11.35) -- cycle;  
  \draw [lightestblue!40,fill=lightestblue!40] (0:0) -- (-11.35,0) -- (-11.35,-11.35) -- (0,-11.35) -- cycle;  
  \draw [lightestred!30,fill=lightestred!30] (0:0) -- (11.35,0) -- (11.35,-11.35) -- (0,-11.35) -- cycle;   
  \foreach \r in {1,...,11} 
  {
\draw[faded] (\r,-11.35) -- (\r,11.35);
  \draw[faded] (-\r,-11.35) -- (-\r,11.35);
  }
  
  \foreach \r in {1,...,5} 
  {
  \draw[faded] (-11.35,2.236*\r) -- (11.35,2.236*\r);
  \draw[faded] (-11.35,-2.236*\r) -- (11.35,-2.236*\r);
  }  
  %%
  \draw (-11.35,0) -- (11.35,0);
  \draw (0,-11.35) -- (0,11.35);
  %%
  %%
  \draw [lightestred] (0,0) circle (3cm);
  \draw [lightestred] (0,0) circle (4.583cm);
  \draw [lightestred] (0,0) circle (7cm);
  \draw [lightestred] (0,0) circle (8.307cm);
  \draw [lightestred] (0,0) circle (11.358cm);

  \draw [lightestred,domain=17.08:72.92] plot ({11.874*cos(\x)}, {11.874*sin(\x)});
  \draw [lightestred,domain=-17.08:-72.92] plot ({11.874*cos(\x)}, {11.874*sin(\x)});
  \draw [lightestred,domain=162.92:107.08] plot ({11.874*cos(\x)}, {11.874*sin(\x)});
  \draw [lightestred,domain=-162.92:-107.08] plot ({11.874*cos(\x)}, {11.874*sin(\x)});

  \draw [lightestred,domain=26.56:63.44] plot ({12.689*cos(\x)}, {12.689*sin(\x)});
  \draw [lightestred,domain=-26.56:-63.44] plot ({12.689*cos(\x)}, {12.689*sin(\x)});
  \draw [lightestred,domain=153.44:116.56] plot ({12.689*cos(\x)}, {12.689*sin(\x)});
  \draw [lightestred,domain=-153.44:-116.56] plot ({12.689*cos(\x)}, {12.689*sin(\x)});
  %%
   \draw [lightestred,domain=38.81:53.19] plot ({14.177*cos(\x)}, {14.177*sin(\x)});
  \draw [lightestred,domain=-38.81:-53.19] plot ({14.177*cos(\x)}, {14.177*sin(\x)});
  \draw [lightestred,domain=143.19:126.81] plot ({14.177*cos(\x)}, {14.177*sin(\x)});
  \draw [lightestred,domain=-143.19:-126.81] plot ({14.177*cos(\x)}, {14.177*sin(\x)});
  %%
  %\draw [lightestred] (0,0) circle (2cm);
  %% 1st quadrant
  \node [p] at (2,0) {}; %% exceptional ?
  \node [i] at (3,0) {}; %% norm = 9=3*3
  \node [i] at (7,0) {}; %% norm = 49=7*7
  \node [p] at (11,0) {}; %% 11 (mod 20)
  \node [p] at (0,1*2.236) {}; %% exceptional ?
  \node [c] at (1,1*2.236) {}; %% norm = 6=2*3
  \node [i] at (2,1*2.236) {}; %% norm = 9=3*3
  \node [c] at (3,1*2.236) {}; %% norm = 14=2*7
  \node [i] at (4,1*2.236) {};  %% norm = 21=3*7
  \node [p] at (6,1*2.236) {};  %% norm = 41 (prime)
  \node [i] at (8,1*2.236) {};  %% norm = 69=3*23  
  \node [c] at (9,1*2.236) {}; %% norm = 86=2*43
  \node [i] at (1,2*2.236) {};  %% norm = 21=3*7 
  \node [p] at (3,2*2.236) {};  %% norm = 29 (prime)
  \node [i] at (7,2*2.236) {};  %% norm = 3*23
  \node [p] at (9,2*2.236) {};  %% norm = 101 (prime)
  \node [i] at (11,2*2.236) {};  %% norm = 141=3*47
  \node [c] at (1,3*2.236) {}; %% norm = 46=2*23
  \node [i] at (2,3*2.236) {};  %% norm = 49=7*7
  \node [p] at (4,3*2.236) {};  %% norm = 61 (prime)
  \node [c] at (7,3*2.236) {}; %% norm = 94=2*47
  \node [p] at (8,3*2.236) {}; %% norm = 109 (prime) 
  \node [c] at (11,3*2.236) {}; %% norm = 166=2*83   
  \node [p] at (3,4*2.236) {};  %% norm = 89 (prime)
  \node [i] at (7,4*2.236) {};  %% norm = 129=3*43
  \node [i] at (9,4*2.236) {};  %% norm = 161=7*23  
  \node [i] at (11,4*2.236) {};  %% norm = 201=3*67    
  \node [i] at (9,4*2.236) {};  %% norm = 161=7*23  
  \node [i] at (2,5*2.236) {};  %% norm = 129=3*43
  \node [c] at (3,5*2.236) {}; %% norm = 134=2*67 
  \node [i] at (4,5*2.236) {};  %% norm = 141=3*47 
  \node [i] at (6,5*2.236) {};  %% norm = 161=7*23
  \node [c] at (9,5*2.236) {}; %% norm = 206=2*103   
  %%
  %% 2nd quadrant
  \node [p] at (-2,0) {}; %% exceptional ?
  \node [i] at (-3,0) {}; %% norm = 9=3*3
  \node [i] at (-7,0) {}; %% norm = 49=7*7
  \node [p] at (-11,0) {}; %% 11 (mod 20)
  %\node [p] at (0,1*2.236) {}; %% exceptional ?
  \node [c] at (-1,1*2.236) {}; %% norm = 6=2*3
  \node [i] at (-2,1*2.236) {}; %% norm = 9=3*3
  \node [c] at (-3,1*2.236) {}; %% norm = 14=2*7
  \node [i] at (-4,1*2.236) {};  %% norm = 21=3*7
  \node [p] at (-6,1*2.236) {};  %% norm = 41 (prime)
  \node [i] at (-8,1*2.236) {};  %% norm = 69=3*23  
  \node [c] at (-9,1*2.236) {}; %% norm = 86=2*43
  \node [i] at (-1,2*2.236) {};  %% norm = 21=3*7 
  \node [p] at (-3,2*2.236) {};  %% norm = 29 (prime)
  \node [i] at (-7,2*2.236) {};  %% norm = 3*23
  \node [p] at (-9,2*2.236) {};  %% norm = 101 (prime)
  \node [i] at (-11,2*2.236) {};  %% norm = 141=3*47
  \node [c] at (-1,3*2.236) {}; %% norm = 46=2*23
  \node [i] at (-2,3*2.236) {};  %% norm = 49=7*7
  \node [p] at (-4,3*2.236) {};  %% norm = 61 (prime)
  \node [c] at (-7,3*2.236) {}; %% norm = 94=2*47
  \node [p] at (-8,3*2.236) {}; %% norm = 109 (prime)
  \node [c] at (-11,3*2.236) {}; %% norm = 166=2*83   
  \node [p] at (-3,4*2.236) {};  %% norm = 89 (prime)
  \node [i] at (-7,4*2.236) {};  %% norm = 129=3*43
  \node [i] at (-9,4*2.236) {};  %% norm = 161=7*23  
  \node [i] at (-11,4*2.236) {};  %% norm = 201=3*67    
  \node [i] at (-9,4*2.236) {};  %% norm = 161=7*23  
  \node [i] at (-2,5*2.236) {};  %% norm = 129=3*43
  \node [c] at (-3,5*2.236) {}; %% norm = 134=2*67 
  \node [i] at (-4,5*2.236) {};  %% norm = 141=3*47 
  \node [i] at (-6,5*2.236) {};  %% norm = 161=7*23
  \node [c] at (-9,5*2.236) {}; %% norm = 206=2*103
  %% 4th quadrant
  %\node [p] at (2,0) {}; %% exceptional ?
  %\node [i] at (3,0) {}; %% norm = 9=3*3
  %\node [i] at (7,0) {}; %% norm = 49=7*7
  %\node [p] at (11,0) {}; %% 11 (mod 20)
  \node [p] at (0,-1*2.236) {}; %% exceptional ?
  \node [c] at (1,-1*2.236) {}; %% norm = 6=2*3
  \node [i] at (2,-1*2.236) {}; %% norm = 9=3*3
  \node [c] at (3,-1*2.236) {}; %% norm = 14=2*7
  \node [i] at (4,-1*2.236) {};  %% norm = 21=3*7
  \node [p] at (6,-1*2.236) {};  %% norm = 41 (prime)
  \node [i] at (8,-1*2.236) {};  %% norm = 69=3*23  
  \node [c] at (9,-1*2.236) {}; %% norm = 86=2*43
  \node [i] at (1,-2*2.236) {};  %% norm = 21=3*7 
  \node [p] at (3,-2*2.236) {};  %% norm = 29 (prime)
  \node [i] at (7,-2*2.236) {};  %% norm = 3*23
  \node [p] at (9,-2*2.236) {};  %% norm = 101 (prime)
  \node [i] at (11,-2*2.236) {};  %% norm = 141=3*47
  \node [c] at (1,-3*2.236) {}; %% norm = 46=2*23
  \node [i] at (2,-3*2.236) {};  %% norm = 49=7*7
  \node [p] at (4,-3*2.236) {};  %% norm = 61 (prime)
  \node [c] at (7,-3*2.236) {}; %% norm = 94=2*47
  \node [p] at (8,-3*2.236) {}; %% norm = 109 (prime)
  \node [c] at (11,-3*2.236) {}; %% norm = 166=2*83   
  \node [p] at (3,-4*2.236) {};  %% norm = 89 (prime)
  \node [i] at (7,-4*2.236) {};  %% norm = 129=3*43
  \node [i] at (9,-4*2.236) {};  %% norm = 161=7*23  
  \node [i] at (11,-4*2.236) {};  %% norm = 201=3*67    
  \node [i] at (9,-4*2.236) {};  %% norm = 161=7*23  
  \node [i] at (2,-5*2.236) {};  %% norm = 129=3*43
  \node [c] at (3,-5*2.236) {}; %% norm = 134=2*67 
  \node [i] at (4,-5*2.236) {};  %% norm = 141=3*47 
  \node [i] at (6,-5*2.236) {};  %% norm = 161=7*23
  \node [c] at (9,-5*2.236) {}; %% norm = 206=2*103   
  %%
  %% 3rd quadrant
  %\node [p] at (-2,0) {}; %% exceptional ?
  %\node [i] at (-3,0) {}; %% norm = 9=3*3
  %\node [i] at (-7,0) {}; %% norm = 49=7*7
  %\node [p] at (-11,0) {}; %% 11 (mod 20)
  %\node [p] at (0,1*2.236) {}; %% exceptional ?
  \node [c] at (-1,-1*2.236) {}; %% norm = 6=2*3
  \node [i] at (-2,-1*2.236) {}; %% norm = 9=3*3
  \node [c] at (-3,-1*2.236) {}; %% norm = 14=2*7
  \node [i] at (-4,-1*2.236) {};  %% norm = 21=3*7
  \node [p] at (-6,-1*2.236) {};  %% norm = 41 (prime)
  \node [i] at (-8,-1*2.236) {};  %% norm = 69=3*23  
  \node [c] at (-9,-1*2.236) {}; %% norm = 86=2*43
  \node [i] at (-1,-2*2.236) {};  %% norm = 21=3*7 
  \node [p] at (-3,-2*2.236) {};  %% norm = 29 (prime)
  \node [i] at (-7,-2*2.236) {};  %% norm = 3*23
  \node [p] at (-9,-2*2.236) {};  %% norm = 101 (prime)
  \node [i] at (-11,-2*2.236) {};  %% norm = 141=3*47
  \node [c] at (-1,-3*2.236) {}; %% norm = 46=2*23
  \node [i] at (-2,-3*2.236) {};  %% norm = 49=7*7
  \node [p] at (-4,-3*2.236) {};  %% norm = 61 (prime)
  \node [c] at (-7,-3*2.236) {}; %% norm = 94=2*47
  \node [p] at (-8,-3*2.236) {}; %% norm = 109 (prime)
  \node [c] at (-11,-3*2.236) {}; %% norm = 166=2*83   
  \node [p] at (-3,-4*2.236) {};  %% norm = 89 (prime)
  \node [i] at (-7,-4*2.236) {};  %% norm = 129=3*43
  \node [i] at (-9,-4*2.236) {};  %% norm = 161=7*23  
  \node [i] at (-11,-4*2.236) {};  %% norm = 201=3*67    
  \node [i] at (-9,-4*2.236) {};  %% norm = 161=7*23  
  \node [i] at (-2,-5*2.236) {};  %% norm = 129=3*43
  \node [c] at (-3,-5*2.236) {}; %% norm = 134=2*67 
  \node [i] at (-4,-5*2.236) {};  %% norm = 141=3*47 
  \node [i] at (-6,-5*2.236) {};  %% norm = 161=7*23
  \node [c] at (-9,-5*2.236) {}; %% norm = 206=2*103
  %%
  \draw [faded] (0,0) circle (1cm);
  \node [p,fill=white] at (1,0) {};
  \node [p,fill=white] at (-1,0) {};
  %%
  \draw [white,fill=white] (-7,11.35) rectangle (7,11.4);  %% need to change
  \draw [white,fill=white] (-7,-11.35) rectangle (7,-11.4);
  \draw [white,fill=white] (-11.35,-7) rectangle (-11.4,7);
  \draw [white,fill=white] (11.35,-7) rectangle (11.4,7);  
  %%
  \end{tikzpicture}
  \]
  \begin{enumerate}
  \item Create an analogous picture for the ring
    $R_{-6}=\Z[\sqrt{-6}]$. First make a blank diagram with the norms
    of the quadratic integers labeled at each correponding lattice
    point.
  \item Find primes $p\in\Z$ that in $R_{-6}$ are inert, split (both
    reducible and irreducible), and ramified. Illustrate this with
    subring lattices. Examples for $R_{-5}$ are shown below.
    \[
    \hspace*{-5mm}
    \begin{tikzpicture}[scale=1.5,yscale=1.2]
      %\begin{tikzpicture}[scale=.9,yscale=1.4,xscale=1.25]
     \tikzstyle{every node}=[font=\small]
      \begin{scope}[shift={(0,0)}]
        \node[Blue] (R) at (0,2) {$\Z[\sqrt{-5}]$};
        \node[Red] (Z) at (0,1) {$\Z$};
        \node[Blue] (11) at (0,0) {$(11)$};
        \draw (11) to (Z); \draw (Z) to (R);
      \end{scope}
      %%
      \begin{scope}[shift={(2.35,0)}]
        \node[Blue] (R) at (0,2) {$\Z[\sqrt{-5}]$};
        \node[Blue] (1-6root-5) at (-1,1) {$(3\!-\!2\sqrt{-5})$};
        \node[Red] (Z) at (0,1) {$\Z$};
        \node[Blue] (1+6root-5) at (1,1) {$(3\!+\!2\sqrt{-5})$};
        \node[Blue] (11) at (0,0) {$(29)$};
        \draw (11) to (Z); \draw (Z) to (R);
        \draw (11) to (1-6root-5); \draw (11) to (1+6root-5); 
        \draw (R) to (1-6root-5); \draw (R) to (1+6root-5); 
      \end{scope}
      %%
      \begin{scope}[shift={(5.95,0)}]
        \node[Blue] (R) at (0,2) {$\Z[\sqrt{-5}]$};
        \node[Blue] (1-6root-5) at (-1,1) {$(3,2\!-\!\sqrt{-5})$};
        \node[Red] (Z) at (0,1) {$\Z$};
        \node[Blue] (1+6root-5) at (1,1) {$(3,2\!+\!\sqrt{-5})$};
        \node[Blue] (11) at (0,0) {$(3)$};
        \draw (11) to (Z); \draw (Z) to (R);
        \draw (11) to (1-6root-5); \draw (11) to (1+6root-5); 
        \draw (R) to (1-6root-5); \draw (R) to (1+6root-5); 
      \end{scope}
      %%
      \begin{scope}[shift={(8.5,0)}]
        \node[Blue] (R) at (0,2) {$\Z[\sqrt{-5}]$};
        \node[Blue] (root-5) at (1,1) {$(\sqrt{-5})$};
        \node[Red] (Z) at (0,1) {$\Z$};
        \node[Blue] (5) at (0,0) {$(5)$};
        \draw (5) to (Z); \draw (Z) to (R);
        \draw (5) to (root-5); \draw (R) to (root-5); 
      \end{scope}
    \end{tikzpicture}
    \]    
  \item Give an elementary characterization of non-prime
    irreducibles in $R_{-6}$. % \medskip
  \end{enumerate}


\item Prove the following basic facts about principal ideal domains (PIDs). \vspace{-1mm}
  \begin{enumerate}[(a)]
  \item The following three conditions are equivalent for nonzero $a,b\in R$: %\vspace{-1mm}
    \begin{enumerate}[(i)]
    \item $a$ and $b$ are associates (that is, $a\mid b$ and $b\mid a$),
    \item $a=bu$ for some unit $u\in R$,
    \item $(a)=(b)$.
    \end{enumerate}
  \item The following three conditions are equivalent for an element $a\in R$: %\vspace{-1mm}
    \begin{enumerate}[(i)]
    \item $a$ is irreducible
    \item the ideal $(a)$ is maximal
    \item the ideal $(a)$ is prime
    \end{enumerate}
  \item Any two nonzero elements $a,b\in R$ have a unique LCM, up to
    associates, which is a generator $m$ of $(a)\cap (b)$.
  \end{enumerate}

  
\item If the quadratic integer ring $R_m$ is a Euclidean domain, then for every nonzero $a,b\in R$, the division algorithm can be used to find $r,q\in R$ such that
  \[
  a=bq+r,\qquad 0\leq N(r)<N(b).
  \]
  \begin{enumerate}

  \item The following visual shows all three ways to write $a=bq+r$
    in the Gaussian integers, for $a=6+3i$ and $b=1+2i$. Carry out the
    division algorithm for $a=9+8i$ and $b=3+i$, and create an analogous
    picture to illustrate it.
    %% Gaussian integers
    \[\hspace*{-10mm}
    \begin{tikzpicture}[scale=1.35]
      \tikzstyle{0} = [circle, draw, black, fill=black,inner sep=0pt, 
        minimum size=2mm]
      \tikzstyle{p} = [circle, draw, faded, fill=faded,inner sep=0pt, 
        minimum size=1.5mm]
      \tikzstyle{B} = [circle, draw, Blue, fill=Blue,inner sep=0pt, 
        minimum size=2mm]
      \tikzstyle{A} = [circle, draw, Red, fill=Red,inner sep=0pt, 
        minimum size=2mm]
      \tikzstyle{p} = [circle, draw, faded, fill=faded,inner sep=0pt, 
        minimum size=1.5mm]
      \tikzstyle{G} = [circle, draw, faded, fill=Blue,inner sep=0pt, 
        minimum size=1.5mm]
      \tikzstyle{every node}=[font=\footnotesize]
      %%
      \draw [dashed,Blue,fill=lightestblue!40] (6,3) circle (2.236cm);
      \draw [dashed,Blue,fill=lightestblue!40] (0,0) circle (2.236cm);
      
      %% positive slope
      \draw [lightestblue, very thin] (-2.5,5) to (-1.75,6.5);  
      \draw [lightestblue, very thin] (-2.5,0) to (.75,6.5);
      \draw [lightestblue, very thin] (-1.25,-2.5) to (3.25,6.5);
      \draw [lightestblue, very thin] (-1.25+2.5,-2.5) to (3.25+2.5,6.5);
      \draw [lightestblue, very thin] (-1.25+2*2.5,-2.5) to (3.25+2*2.5,6.5);
      \draw [lightestblue, very thin] (-1.25+3*2.5,-2.5) to (8.5,2);
      %%
      %% negative slope
      \draw [lightestblue, very thin] (-2.5,-1.25) to (0,-2.5);
      \draw [lightestblue, very thin] (-2.5,1.25) to (5,-2.5);
      \draw [lightestblue, very thin] (-2.5,1.25+2.5) to (8.5,-1.75);
      \draw [lightestblue, very thin] (-2.5,1.25+2*2.5) to (8.5,-1.75+2.5);
  \draw [lightestblue, very thin] (2,6.5) to (8.5,-1.75+2*2.5);
  \draw [lightestblue, very thin] (7,6.5) to (8.5,5.75);
  
  \foreach \x in {-2,...,8} 
  {
  \foreach \y in {-2,...,6} 
  {
  \node [p] at (\x,\y) {};
  }}
    \node [0] (0) at (0,0) {};
    \node [B] (b) at (1,2) {};
    \node [A] (a) at (6,3) {};
    \node [B] (ib) at (-2,1) {};
    \node [B] (-b) at (-1,-2) {};
    \node [B] (-ib) at (2,-1) {};
    \node [B] at (3,1) {};
    \node [B] at (7,4) {}; \node [B] at (6,2) {}; \node [B] at (4,3) {};
    %%
    \node [G] at (-2,6) {}; \node [G] at (3,6) {}; \node [G] at (8,6) {};
    \node [G] at (0,5) {}; \node [G] at (5,5) {};
    \node [G] at (-1,3) {}; \node [G] at (2,4) {};
    \node [G] at (5,0) {}; \node [G] at (8,1) {};
    \node [G] at (4,-2) {}; \node [G] at (7,-1) {}; 
    %%
    \draw [b] (0) to (b);
    \draw [b,dotted] (0) to (ib);
    \draw [b,dotted] (0) to (-b);
    \draw [b,dotted] (0) to (-ib);
    \draw [dashed,Blue] (0,0) circle (2.236cm);
    \draw [dashed,Blue] (6,3) circle (2.236cm);
    \draw [b] (b) to (3,1); 
    \draw [b] (3,1) to (4,3);
    \draw [b] (4,3) to (6,2);
    \draw [b] (6,2) to (7,4);
  %%
    \draw[->] (-0,-2.6) to (0,6.6);
    \node at (-.4,6.5) {\normalsize $\Im$};
    \draw[->] (-2.6,0) -- (8.6,0);
    \node at (8.5,-.4) {\normalsize $\Re$};
  %%
  \node [Blue] at (.85,2.4) {\small $b=1+2i$};
  \node [Blue] at (-2,1.5) {$ib$};
  \node [Blue] at (-1.3,-1.5) {$-b$};
  \node [Blue] at (2.1,-1.5) {$-ib$};
  \node [Blue] at (3,.7) {$b-ib$};
  \node [Blue] at (4.3,3.4) {$2b-ib$};
  \node [Blue] at (6.3,1.6) {$2b-2ib$};
  \node [Blue] at (6.8,4.4) {$3b-2ib$};
  \node [Red] at (5.9,3.4) {\small $a$};
  \draw [g,thick] (4,3) to (6,3);
  \draw [g,thick] (6,2) to (6,3);
  \draw [g,thick] (7,4) to (6,3);
   \node [A] at (6,3) {};
  %%
  \end{tikzpicture}
  \]
%  \caption{An example of the division algorithm in the Gaussian integers, using $\Alert{a=6+3i}$ and $\Balert{b=1+2i}$.}
  

\item The ring $R_{-5}=\Z[\sqrt{-5}]$ is \emph{not} a Euclidean domain, since the division algorithm fails for $a=5$ and $b=2+\sqrt{-5}$, as demonstrated by the following visual. Find an $a$ and $b$ in $R_{-6}=\Z[\sqrt{-6}]$ that confirms that it too is not Euclidean, and construct an analogous visual.
  \[
  \begin{tikzpicture}[scale=1]
  \tikzstyle{0} = [circle, draw, black, fill=black,inner sep=0pt, 
    minimum size=2mm]
  \tikzstyle{p} = [circle, draw, faded, fill=faded,inner sep=0pt, 
    minimum size=1.5mm]
  \tikzstyle{B} = [circle, draw, Blue, fill=Blue,inner sep=0pt, 
    minimum size=2mm]
  \tikzstyle{A} = [circle, draw, Red, fill=Red,inner sep=0pt, 
    minimum size=2mm]
  \tikzstyle{p} = [circle, draw, faded, fill=faded,inner sep=0pt, 
    minimum size=1.5mm]
  \tikzstyle{G} = [circle, draw, faded, fill=Blue,inner sep=0pt, 
    minimum size=1.65mm]

  \draw [dashed,Blue,fill=lightestblue!40] (5,0) circle (3cm);
    \tikzstyle{every node}=[font=\scriptsize]
  \draw [faded] (0,0) circle (1cm);    
    
  %%    
  \foreach \i in {-2,...,4} 
  {
  \draw[faded] (-6.35,2.236*\i) -- (9.35,2.236*\i);
  }  
  \foreach \i in {-6,...,9} 
  {
  \draw[faded] (\i,-4.8) -- (\i,9.25);
  }  
  \draw (-6.35,0) -- (9.35,0);
  \draw (0,-4.8) -- (0,9.25);
  %%
  %% Positively sloped lines
    \draw[domain=-6:-0.73, smooth, variable=\x, lightestblue, very thin] plot ({\x}, {1.118*(\x+9)+0});
    \draw[domain=-4.29:8.27, smooth, variable=\x, lightestblue, very thin] plot ({\x}, {1.118*(\x-0)+0});
    \draw[domain=4.7:9.35, smooth, variable=\x, lightestblue, very thin] plot ({\x}, {1.118*(\x-9)+0});
    
  %% Negatively sloped lines
    \draw[domain=7.65:9.35, smooth, variable=\x, lightestblue, very thin] plot ({\x}, {-.894*(\x-18)+0});
  \draw[domain=3.15:9.35, smooth, variable=\x, lightestblue, very thin] plot ({\x}, {-.894*(\x-13.5)+0});
  \draw[domain=-1.35:9.35, smooth, variable=\x, lightestblue, very thin] plot ({\x}, {-.894*(\x-9)+0});
  \draw[domain=-5.85:9.35, smooth, variable=\x, lightestblue, very thin] plot ({\x}, {-.894*(\x-4.5)+0});
  \draw[domain=-6.35:5.37, smooth, variable=\x, lightestblue, very thin] plot ({\x}, {-.894*\x});
  \draw[domain=-6.35:0.87, smooth, variable=\x, lightestblue, very thin] plot ({\x}, {-.894*(\x+4.5)});
    \draw[domain=-6.35:-3.63, smooth, variable=\x, lightestblue, very thin] plot ({\x}, {-.894*(\x+9)});
  %%
  \node [Blue] at (1.7,2.9) {\small $b\!=\!2\!+\!\sqrt{-5}$};
  \node [Blue] at (-4,2*2.236) {\small $\sqrt{-5}b$};
  %\node [Blue] at (-2,1.5) {$ib$};
  \node [A] at (5,0) {};
  \node [B] at (2,2.236) {};
  \node [Red] at (5.4,.6) {\small $a=5$};
  \node [G] at (-1,4*2.236) {};
  \node [G] at (8,4*2.236) {};
  \node [G] at (-3,3*2.236) {};
  \node [G] at (6,3*2.236) {};
  \node [B] at (-5,2*2.236) {};
  \node [G] at (4,2*2.236) {};
  \node [G] at (0,0) {};
  \node [G] at (9,0) {};
  \node [G] at (-2,-2.236) {};
  \node [G] at (7,-2.236) {};
  \node [G] at (-4,-2*2.236) {};
  \node [G] at (5,-2*2.236) {};
  \draw [b] (0,0) to (2,2.236);
  \draw [b] (0,0) to (-5,2*2.236);
  %%
  \end{tikzpicture}
  \]
\end{enumerate}

  
\end{enumerate}
  
  %%----------------------------------------------------------
  
\end{document}


