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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due April 21 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4130 $|$ Visual Algebra $|$
    Spring 2023 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}

\item The Galois group of $x^n-1$ naturally acts on the $n^{\rm th}$ roots of unity; this is shown below for $n=3,\dots,8$. The primitive roots are highlighted.
  \[
  \hspace*{-5mm}
    \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, 
      minimum size=4.5mm]
       \tikzstyle{v-f} = [circle, draw, grey, fill=lightgrey,inner sep=0pt, 
    minimum size=4.5mm]
    \tikzstyle{b} = [draw, ultra thick,Blue,-stealth]
    \tikzstyle{bb} = [draw, ultra thick,Blue]
    \tikzstyle{b-f} = [draw, ultra thick, lightblue,-stealth]
    \tikzstyle{bb-f} = [draw, ultra thick, lightblue]
    \tikzstyle{gg} = [draw, ultra thick, Green]
    \tikzstyle{gg-f} = [draw, ultra thick, lightgreen]
    \tikzstyle{g-f} = [draw, ultra thick, lightgreen]
    \tikzstyle{every node}=[font=\scriptsize]
    \tikzset{every loop/.style={min distance=5mm,looseness=10}}
    \begin{tikzpicture}[scale=1.5]
    %%
      \begin{scope}[shift={(0,0)}]  
        \draw [faded] (0,-1.3) to (0,1.3);
        \draw [faded] (-1.55,0) to (1.65,0);
        \draw[faded] (0,0) circle [radius=1cm];
        \node (1) at (0:1) [v-f] {$1$};
        \node (z) at (120:1) [v] {$\zeta$};
        \node (z2) at (240:1) [v] {$\zeta^2$};
        %%
        \path (1) edge [b-f,loop right,-stealth] (1);
        \draw[bb] (z) to (z2); 
      \end{scope}
      %%
      \begin{scope}[shift={(3.5,0)}]  
        \draw [faded] (0,-1.3) to (0,1.3);
        \draw [faded] (-1.65,0) to (1.65,0);
        \draw[faded] (0,0) circle [radius=1cm];
        \node (1) at (0:1) [v-f] {$1$};
        \node (z) at (90:1) [v] {$\zeta$};
        \node (z2) at (180:1) [v-f] {$\zeta^2$};
        \node (z3) at (270:1) [v] {$\zeta^3$};
        %%
        \path (1) edge [b-f,loop right,-stealth] (1);
        \path (z2) edge [b-f,loop left,-stealth] (z2);
        \draw[bb] (z) to (z3); 
      \end{scope}
      %%
      \begin{scope}[shift={(7,0)}]
        \draw [faded] (0,-1.3) to (0,1.3);
        \draw [faded] (-1.55,0) to (1.6,0);
        \draw[faded] (0,0) circle [radius=1cm];
        \node (1) at (0:1) [v-f] {\small $1$};
        \node (z) at (72:1) [v] {$\zeta$};
        \node (z2) at (144:1) [v] {$\zeta^2$};
        \node (z3) at (216:1) [v] {$\zeta^3$};
        \node (z4) at (288:1) [v] {$\zeta^4$};
        %%
        \path (1) edge [b-f,loop right,-stealth] (1);
        \draw[b] (z) to (z3); \draw[b] (z3) to (z4);
        \draw[b] (z4) to (z2); \draw[b] (z2) to (z);
      \end{scope}
      %%
      \begin{scope}[shift={(0,-3)}]
        \draw [faded] (0,-1.3) to (0,1.3);
        \draw [faded] (-1.65,0) to (1.65,0);
        \draw[faded] (0,0) circle [radius=1cm];
        \node (1) at (0:1) [v-f] {\small $1$};
        \node (z) at (60:1) [v] {$\zeta$};
        \node (z2) at (120:1) [v-f] {$\zeta^2$};
        \node (z3) at (180:1) [v-f] {$\zeta^3$};
        \node (z4) at (240:1) [v-f] {$\zeta^4$};
        \node (z5) at (300:1) [v] {$\zeta^5$};
        %%
        \path (1) edge [b-f,loop right,-stealth] (1);
        \path (z3) edge [b-f,loop left,-stealth] (z3);
        \draw[bb] (z) to (z5); \draw[bb-f] (z2) to (z4);
      \end{scope}
    %%
      \begin{scope}[shift={(3.5,-3)}]
        \draw [faded,very thin] (0,-1.3) to (0,1.3);
        \draw [faded,very thin] (-1.55,0) to (1.65,0);
        \draw[faded,very thin] (0,0) circle [radius=1cm];
        \node (1) at (0:1) [v-f] {\small $1$};
        \node (z) at (51.4:1) [v] {$\zeta$};
        \node (z2) at (102.8:1) [v] {$\zeta^2$};
        \node (z3) at (154.3:1) [v] {$\zeta^3$};
        \node (z4) at (205.7:1) [v] {$\zeta^4$};
        \node (z5) at (257.1:1) [v] {$\zeta^5$};
        \node (z6) at (308.6:1) [v] {$\zeta^6$};
        %%
        \path (1) edge [b-f,loop right,-stealth] (1);
        \draw[b,ultra thick] (z) to (z3); \draw[b,ultra thick] (z3) to (z2); 
        \draw[b,ultra thick] (z2) to (z6); \draw[b,ultra thick] (z6) to (z4); 
        \draw[b,ultra thick] (z4) to (z5); \draw[b,ultra thick] (z5) to (z); 
      \end{scope}
      %%
      \begin{scope}[shift={(7,-3)}]
        \draw [faded,very thin] (0,-1.3) to (0,1.3);
        \draw [faded,very thin] (-1.65,0) to (1.65,0);
        \draw[faded,very thin] (0,0) circle [radius=1cm];
        \node (1) at (0:1) [v-f] {\small $1$};
        \node (z) at (45:1) [v] {$\zeta$};
        \node (z2) at (90:1) [v-f] {$\zeta^2$};
        \node (z3) at (135:1) [v] {$\zeta^3$};
        \node (z4) at (180:1) [v-f] {$\zeta^4$};
        \node (z5) at (225:1) [v] {$\zeta^5$};
        \node (z6) at (270:1) [v-f] {$\zeta^6$};
        \node (z7) at (313:1) [v] {$\zeta^7$};
        %%
        \path (1) edge [b-f,loop below,-stealth] (1);
        \path (z4) edge [b-f,loop below,-stealth] (z4);
        \path (z2) edge [g-f,loop above,-stealth] (z2);
        \path (z6) edge [g-f,loop below,-stealth] (z6);
        \path (1) edge [g-f,loop above,-stealth] (1);
        \path (z4) edge [g-f,loop above,-stealth] (z4);
        %%
        \draw[bb,ultra thick] (z) to (z7); \draw[bb-f] (z2) to (z6); \draw[bb,ultra thick] (z3) to (z5);
        \draw[gg,ultra thick] (z) to (z5); \draw[gg,ultra thick] (z3) to (z7);
      \end{scope}
      \end{tikzpicture}
    \]

    \vspace{-9mm}
    
      \begin{enumerate}
      \item Construct analogous diagrams for $n=9$, $10$, and $16$.
      \item For each of these three, write the splitting field of
        $x^n-1$ with elements that do not involve $\zeta$. If possible, write
        the generating automorphism(s) of $\Gal(x^n-1)$ in terms of
        them.  For example, $\Q(\zeta_8)=\Q(\sqrt{2},i)$, and
        $\Gal(x^8-1)=\<\sigma,\tau\>\cong \Z_8^\times\cong V_4$, where
        \[
        \left\{\begin{array}{ll}
        \sigma\colon\sqrt{2}\longmapsto\sqrt{2} \\
        \sigma\colon\;\;\;i\longmapsto -i
        \end{array}\right.\qquad\qquad
        %%
        \left\{\begin{array}{ll}
        \tau\colon\sqrt{2}\longmapsto-\sqrt{2} \\
        \tau\colon\;\;\;i\longmapsto i 
        \end{array}\right.
        \]
        \vspace{-3.5mm}
        
      \item Draw the subgroup lattice of $\Gal(x^n-1)$ and the subfield
        lattice of $\Q(\zeta)$.
      \end{enumerate}
        
  
\item Consider the following polynomial that is irreducible over $\Q$:
  \[
  f(x)=x^4-x^2-5=\Big(x^2-\sqrt{\tfrac{1}{2}+\tfrac{\sqrt{21}}{2}}\Big)
  \Big(x^2-\sqrt{\tfrac{1}{2}-\tfrac{\sqrt{21}}{2}}\Big).
  \]
  Denote its roots by $r_1,r_2,r_3,r_4$, its splitting field
  by $K=\Q(r_1,r_2,r_3,r_4)$, and its Galois group over $\Q$ by $G=\Gal(f(x))$.
  \begin{enumerate}
  \item Since $f(x)$ is irreducible, $[\Q(r_i):\Q]=4$. Use this
    to find a lower bound on $[K:\Q]$.
  \item By the tower law, $[K:\Q]=[\Q(r_1,r_2,r_3,r_4):\Q]$ is equal to
    \[
      [\Q(r_1,r_2,r_3,r_4):\Q(r_1,r_2,r_3)]\cdot
      [\Q(r_1,r_2,r_3):\Q(r_1,r_2)]\cdot
      [\Q(r_1,r_2):\Q(r_1)]\cdot
      [\Q(r_1):\Q].
    \]
    Use this to find an upper bound on $[K:\Q]$.
  \item Find $[K:\Q]=|\Gal(f(x)|$, and then the Galois group
    by process of elimination.
  \end{enumerate}



\item Suppose $\alpha\neq 0$ is algebraic over $\Q$ and
  $[\Q(\alpha):\Q]$ is odd.
  \begin{enumerate}
  \item Show that $\Q(\alpha)=\Q(\alpha+\alpha^{-1})$.
  \item Give an example to show how this can fail if $[\Q(\alpha):\Q]$
    is even. 
  \item Repeat the previous two parts, but for the subfield
    $\Q(\alpha^2-\alpha)$. 
  \end{enumerate}


\end{enumerate}

\end{document}



