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\pagestyle{fancy} \lhead{{\sf Homework 2 $&#124;$ Due February 3 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4130/6130 $&#124;$ Visual Algebra $&#124;$
 Spring 2023 $&#124;$ 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,ドル
 \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,ドル 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,ドル 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) {2ドルb-ib$};
 \node [Blue] at (6.3,1.6) {2ドルb-2ib$};
 \node [Blue] at (6.8,4.4) {3ドルb-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},ドル 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}
 
 %%----------------------------------------------------------
 
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