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\pagestyle{fancy} \lhead{{\sf Homework 13 $|$ Due December 2 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8510 $|$ Abstract Algebra $|$
    Fall 2022 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}

\item A picture illustrating the quadratic integers $R_{-5}=\big\{a+b\sqrt{-5}\mid
  a,b\in\Z\big\}$ as a subring of $\C$ is shown below, with the
  primes in black, and non-prime irreducibles in red.
\[
  \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]    
  %%    
  \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] (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 [p] at (1,1*2.236) {}; %% norm = 6=2*3
  \node [i] at (2,1*2.236) {}; %% norm = 9=3*3
  \node [p] 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 [p] 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 [p] 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 [p] at (7,3*2.236) {}; %% norm = 94=2*47
  \node [p] at (8,3*2.236) {}; %% norm = 99=9*11 
  \node [p] 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 [p] 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 [p] 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 [i] at (0,1*2.236) {}; %% exceptional
  \node [p] at (-1,1*2.236) {}; %% norm = 6=2*3
  \node [i] at (-2,1*2.236) {}; %% norm = 9=3*3
  \node [p] 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 [p] 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 [p] 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 [p] at (-7,3*2.236) {}; %% norm = 94=2*47
  \node [p] at (-8,3*2.236) {}; %% norm = 99=9*11 
  \node [p] 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 [p] 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 [p] at (-9,5*2.236) {}; %% norm = 206=2*103
  %% 4th quadrant
  %\node [i] 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 [p] at (1,-1*2.236) {}; %% norm = 6=2*3
  \node [i] at (2,-1*2.236) {}; %% norm = 9=3*3
  \node [p] 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 [p] 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 [p] 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 [p] at (7,-3*2.236) {}; %% norm = 94=2*47
  \node [p] at (8,-3*2.236) {}; %% norm = 99=9*11 
  \node [p] 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 [p] 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 [p] at (9,-5*2.236) {}; %% norm = 206=2*103   
  %%
  %% 3rd quadrant
  %\node [i] 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 [i] at (0,1*2.236) {}; %% exceptional
  \node [p] at (-1,-1*2.236) {}; %% norm = 6=2*3
  \node [i] at (-2,-1*2.236) {}; %% norm = 9=3*3
  \node [p] 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 [p] 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 [p] 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 [p] at (-7,-3*2.236) {}; %% norm = 94=2*47
  \node [p] at (-8,-3*2.236) {}; %% norm = 99=9*11 
  \node [p] 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 [p] 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 [p] 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); 
  \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 $R_{-6}$. Start with
    a blank diagram with the norms of the quadratic integers labeled
    at each correponding lattice point.
  \item Give an elementary characterization of non-prime
    irreducibles. \medskip
  \end{enumerate}


  
%%--------------------------------------------------------------------- 
  
\item Consider the quadratic field $K=\Q(\sqrt{m})$ and an odd prime
  $p\in\Z$.
  \begin{enumerate}
  \item Show that if $p\mid m$, then $p$ is ramified in $R_m$, by establishing
    \[
    (p)=\big(p,\sqrt{m}\big)^2.
    \]
  \item Show that if $p\nmid m$ and $m\equiv n^2\pmod{p}$, then $p$
    splits in $R_m$, via
    \[
    (p)=\big(p,n+\sqrt{m}\big)\big(p,n-\sqrt{m}\big).
    \]
  \item Show that if $p\nmid m$ and $m$ is not a quadradic residue mod
    $p$, then $R_m/(p)$ is a field, and hence $p$ is inert in $R_m$.
  \end{enumerate}

%%--------------------------------------------------------------------- 
  
\item Prove that if $m=-3$, $-7$, or $-11$, then $R_m$ is Euclidean
  with $d(r)=|N(r)|$ for all nonzero $r\in R_m$. [\emph{Hint}: Mimic the
    proof of the same result for $m=-2$, $-1$, $2$, and $3$, but
    choose $d\in\Z$ nearest to $2t$ and then $c\in\Z$ so that $c$ is
    as near to $2s$ as possible with $c\equiv d\pmod 2$, then set
    $q=(c+d\sqrt{m})/2$.] \\

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

\item Suppose $P\neq 0$ is a prime ideal in the ring $R_m$ of
  quadratic integers.
  \begin{enumerate}
  \item Show that $P\cap \Z$ is a prime ideal in $\Z$, so $P\cap\Z=(p)$
    for some prime $p$ in $\Z$.
  \item Set $I=pR_m\subseteq P$ and form the quotient ring $R/I$. Show
    that $R/I$, as an additive group, is generated by two elements of finite
    order; hence $R/I$ is finite. 
  \item Show that there is an epimorphism $R/I\onto R/P$ and conclude
    that $R/P$ is finite.
  \item Conclude that every prime ideal in $R_m$ is maximal. \\
  \end{enumerate}

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

\item An element $e\in R$ is called an \emph{idempotent} if $e^2=e$, and two
  nonzero idempotents $e_1,e_2$ are called an \emph{orthogonal pair} if
  $e_1+e_2=1$ and $e_1e_2=0$.
  \begin{enumerate}
  \item Show that the following are equivalent:
    \begin{enumerate}[(i)]
    \item $R$ contains an idempotent different from 0 and 1.
    \item $R$ contains an orthogonal pair of idempotents.
    \item $R\cong R_1\times R_2$ for some rings $R_1$ and $R_2$.
    \end{enumerate}
  \item Give an example of a non-orthogonal pair of distinct idemponents.
  \item Find all idempotents in the ring $\Z/20\Z$.
  \end{enumerate}


\end{enumerate}

\end{document}
