\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym}
\usepackage[all]{xy}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{patterns}
\usetikzlibrary{shapes.geometric}
\usepackage{tkz-graph}
\usepackage{tikz-cd}
\usepackage{enumerate}
\usepackage{multicol}
\usepackage{url}
\usepackage{bm} 

\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in


% Useful math macros
\def\ds{\displaystyle}
\def\bold{\bm}     % Change ``bm'' to ''mathbf'' for upright boldface.

\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}
\newcommand{\vvvv}[4]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \end{bmatrix}}
\newcommand{\vvvvv}[5]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \\ #5 \end{bmatrix}}
\def\<{\langle}
\def\>{\rangle}
\newcommand{\ceil}[1] {\left\lceil #1 \right\rceil}
\newcommand{\floor}[1] {\left\lfloor #1 \right\rfloor}
\def\normal{\lhd}
\def\normaleq{\unlhd}
\def\nnormal{\ntriangleleft}
\def\nnormaleq{\ntrianglelefteq}
\def\Dot{\bm{.}}

%\def\subgroup{\leqslant}
%\def\nsubgroup{\nleqslant}
%\def\normal{\leqslant}
%\def\nnormal{\nleqslant}

\newcommand{\rddots}{\rotatebox{90}{$\ddots$}}
\def\setmin{-}

\def\Sum{\sum\limits}
\def\Prod{\prod\limits}
\def\Cap{\mathop\cap\limits}
\def\Cup{\mathop\cup\limits}
\def\Times{\mathop\times\limits}
\def\Oplus{\mathop\oplus\limits}
\def\Lim{\mathop\lim\limits}

\def\And{\wedge}
\def\Or{\vee}
\def\Not{\neg}

\def\longto{\longrightarrow}
\def\into{\hookrightarrow}
\def\longinto{\longhookrightarrow}
\def\onto{\twoheadrightarrow}


\def\A{\mathbb{A}}
\def\C{\mathbb{C}}
\def\F{\mathbb{F}}
\def\HH{\mathbb{H}}
\def\N{\mathbb{N}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\Z{\mathbb{Z}}


\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\Lcm}{lcm}
\DeclareMathOperator{\del}{\nabla}
\DeclareMathOperator{\Trace}{tr}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\Image}{Im}
\DeclareMathOperator\Perm{Perm}
\DeclareMathOperator\Char{char}
\DeclareMathOperator\Aut{Aut}
\DeclareMathOperator\Inn{Inn}
\DeclareMathOperator\Out{Out}
\DeclareMathOperator\orb{orb}
\DeclareMathOperator\Orb{Orb}
\DeclareMathOperator\Stab{Stab}
\DeclareMathOperator\stab{stab}
\DeclareMathOperator\Fix{Fix}
\DeclareMathOperator\fix{fix}
\DeclareMathOperator\Syl{Syl}
\DeclareMathOperator\Pre{Pre}
\DeclareMathOperator\Ker{Ker}
\DeclareMathOperator\Sym{Sym}
\DeclareMathOperator\Mat{Mat}
\DeclareMathOperator{\cl}{cl}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator\PGL{PGL}
\DeclareMathOperator\PSL{PSL}
\DeclareMathOperator\PSU{PSU}
\DeclareMathOperator\PSP{PsP}
\DeclareMathOperator\Heis{Heis}
\DeclareMathOperator\Pauli{Pauli}
\DeclareMathOperator\Dih{D}
\DeclareMathOperator\Dic{Dic}
\DeclareMathOperator\DQ{DQ}
\DeclareMathOperator\SD{SD}
\DeclareMathOperator\SA{SA}
\DeclareMathOperator\AGL{AGL}
\DeclareMathOperator\Gal{Gal}
\DeclareMathOperator\Deg{Deg}

\DeclareMathOperator\Rect{\mathbf{Rect}}
\DeclareMathOperator\Tri{\mathbf{Tri}}
\DeclareMathOperator\Coin{\mathbf{Coin}}
\DeclareMathOperator\BOX{\mathbf{Box}}
\DeclareMathOperator\Light{\mathbf{Light}}
\DeclareMathOperator\Sq{\mathbf{Sq}}

\DeclareMathOperator\Frieze{\mathbf{Frz}}

% Colors 
\usepackage{xcolor}
\usepackage{color}
\definecolor{darkblue}{rgb}{0, 0, .6}
\definecolor{grey}{rgb}{.7, .7, .7}
\newcommand{\balert}[1]{\textcolor{blue}{#1}}
\newcommand{\galert}[1]{\textcolor{darkgreen}{#1}}
\newcommand{\Palert}[1]{\textcolor{amethyst}{#1}}
\newcommand{\Alert}[1]{\textcolor{Red}{#1}}
\newcommand{\Balert}[1]{\textcolor{Blue}{#1}}
\newcommand{\Galert}[1]{\textcolor{darkgreen}{#1}}


%\hypersetup{colorlinks=true,linkcolor=darkblue,anchorcolor=darkblue,
%  citecolor=darkblue,pagecolor=darkblue,urlcolor=darkblue,
%  pdftitle={},pdfauthor={}
%}                               %% THIS CAUSES COMPILER ERROR
\colorlet{lightred}{red!30!white}
\colorlet{lightblue}{blue!30!white}
\colorlet{lightyellow}{yellow!30!white}
\colorlet{keylime}{green!10!white}
\colorlet{darkgreen}{green!50!black}
\colorlet{midgreen}{green!70!black}
\colorlet{lightgreen}{darkgreen!30!white}
\colorlet{lightpurple}{violet!60!white}
\colorlet{darkgrey}{black!70}
\colorlet{midgrey}{black!50}
\definecolor{Red}{rgb}{.9,0,0}
\definecolor{Blue}{rgb}{0,0,.9}
\colorlet{Green}{green!50!black}
\definecolor{Gold}{rgb}{.83,.67,.08}

\definecolor{conj-purple}{HTML}{E4B7FF}
\definecolor{conj-orange}{HTML}{FFD4B7}

\colorlet{lred}{red!50!white}
\definecolor{lblue}{rgb}{0.60, 0.75, 1}
\colorlet{lgreen}{green!40!white}
\colorlet{lyellow}{yellow!30!white}
%\definecolor{lpurple}{rgb}{0.6 0.35 0.8}
\definecolor{lpurple}{rgb}{0.7 0.45 0.9}
\colorlet{lorange}{orange!50!white}
\definecolor{lightgrey}{rgb}{.85, .85, .85}
\definecolor{faded}{rgb}{.75,.75,.75}
\definecolor{midgrey}{rgb}{.5,.5,.5}
%\definecolor{midgrey}{rgb}{.35, .35, .35}
%\definecolor{amethyst}{rgb}{0.6 0.4 0.8}
\definecolor{amethyst}{rgb}{0.54 0.17 0.89}
\definecolor{amber}{rgb}{1.0 0.49 0.0}
\definecolor{turq}{rgb}{0.12 0.69 0.71}

\colorlet{lightestgray}{gray!15!white}

\colorlet{lightestblue}{blue!15!white}
\colorlet{midblue}{blue!45!white}

\colorlet{lightestred}{red!15!white}
\colorlet{midred}{red!45!white}
\colorlet{lightestgreen}{darkgreen!15!white}
\colorlet{lightestpurple}{amethyst!20!white}



% COLORS FOR MULTIPLICATION TABLES  
\colorlet{tred}{red!70!white}
\colorlet{tblue}{blue!60!white}     %(higher number is more faded)
\colorlet{tgreen}{green!75!black}   %(higher number is brighter)
\colorlet{tyellow}{yellow!60!white}
\colorlet{tlime}{green!80!white}
\colorlet{tpink}{pink}
\colorlet{tpurple}{lpurple}
\colorlet{torange}{orange}

\newcommand*{\n}{5}   % counter for Cayley tables

%% CAYLEY DIAGRAM GLOBAL DEFAULTS
\tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=6mm]
\tikzstyle{r} = [draw, very thick, Red, -stealth]        % Red --->
\tikzstyle{b} = [draw, very thick, Blue, -stealth]       % Blue --->
\tikzstyle{g} = [draw, very thick, darkgreen, -stealth]  % Green --->
\tikzstyle{p} = [draw, very thick, amethyst, -stealth]     % Purple --->
\tikzstyle{rr} = [draw, very thick, Red]                 % Red ----
\tikzstyle{bb} = [draw, very thick, Blue]                % Blue ----
\tikzstyle{gg} = [draw, very thick, darkgreen]           % Green ----
\tikzstyle{oo} = [draw, very thick, orange]           % Purple ----
\tikzstyle{pp} = [draw, very thick, amethyst]           % Purple ----
\tikzstyle{tt} = [draw, very thick, turq]           % Turquoise ----
\tikzstyle{yy} = [draw, very thick, Gold]           % Gold ----

\tikzstyle{R} = [draw, very thick, Red, -stealth']        % Red --->
\tikzstyle{B} = [draw, very thick, Blue, -stealth']       % Blue --->
\tikzstyle{G} = [draw, very thick, darkgreen, -stealth']  % Green --->

\tikzset{r/.style={draw, very thick, Red, -stealth}}        % Red --->
\tikzset{R/.style={draw, very thick, Red, -stealth'}}        % Red --->

\tikzset{v/.style={circle, draw, fill=lightgrey,inner sep=0pt, 
  minimum size=6mm}}

\renewcommand\labelitemi{\rule[0.12em]{0.4em}{0.4em}}
\renewcommand\labelitemii{\normalfont\bfseries \rule[0.15em]{0.3em}{0.3em}}
\renewcommand\labelitemiii{\textasteriskcentered}
\renewcommand\labelitemiv{\textperiodcentered}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due April 7 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4130 $|$ Visual Algebra $|$
    Spring 2023 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-6mm}

\begin{enumerate}


\item The splitting field of $f(x)=x^8-2$ over $\Q$ is
  $\Q(\sqrt[8]{2},\zeta)=\Q(\sqrt[8]{2},i)$, where $\zeta=e^{2\pi
  i/8}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$, a primitive $8^{\rm
  th}$ root of unity. The Galois group is generated by the
  automorphisms
  \[
  \left\{\begin{array}{ll}
  \rho\colon\sqrt[8]{2}\longmapsto\zeta\sqrt[8]{2} \\
  \rho\colon\;\;\;\zeta\longmapsto\zeta
  \end{array}\right.\qquad\qquad
  \left\{\begin{array}{ll}
  \sigma\colon\sqrt[8]{2}\longmapsto\sqrt[8]{2} \\
  \sigma\colon\;\;\;\, i\longmapsto -i\,,
  \end{array}\right.
  \]
  and is isomorphic to the semidihedral group
  \[
  \SD_8=\<\rho,\sigma\mid\rho^8=1,\;\sigma^2=1,\;\sigma\rho\sigma=\rho^3\>.
  \]
  An action graph of $\Gal(x^8-2)$ acting on the roots is shown below,
  where $\theta=\sqrt[8]{2}$.
  \[
  \begin{tikzpicture}[scale=2.25]
    \begin{scope}[shift={(3,0)}]
      \tikzset{every loop/.style={min distance=5mm,looseness=10}}
      \tikzstyle{every node}=[font=\tiny]
      \draw [faded] (0,-1.4) to (0,1.4);
      \draw [faded] (-1.7,0) to (1.7,0);
      \draw[faded] (0,0) circle [radius=1cm];
      \node (1) at (0:1.091) [v] {$\theta$};
      \node (r) at (45:1.091) [v] {$\zeta\theta$};
      \node (r2) at (90:1.091) [v] {$\zeta^2\theta$};
      \node (r3) at (135:1.091) [v] {$\zeta^3\theta$};
      \node (r4) at (180:1.091) [v] {$\zeta^4\theta$};
      \node (r5) at (225:1.091) [v] {$\zeta^5\theta$};
      \node (r6) at (270:1.091) [v] {$\zeta^6\theta$};
      \node (r7) at (315:1.091) [v] {$\zeta^7\theta$};
      \draw [r] (1) to (r); \draw [r] (r) to (r6);
      \draw [r] (r6) to (r7); \draw [r] (r7) to (r4);
      \draw [r] (r4) to (r5); \draw [r] (r5) to (r2);
      \draw [r] (r2) to (r3); \draw [r] (r3) to (1);
      \path (1) edge [b,loop right,-stealth] (1);
      \path (r4) edge [b,loop left,-stealth] (r4);
      \draw [bb] (r) to (r7);  \draw [bb] (r2) to (r6);
      \draw [bb] (r3) to (r5);
      %%
    \end{scope}
  \end{tikzpicture}
  \]

  \begin{enumerate}
  \item For each subgroup $H\leq\Gal(x^8-2)$, find the largest
    subgroup of $\Q(\sqrt[8]{2},i)$ fixed by $H$, and write it in the
    corresponding place on the subfield lattice on the
    right. %\smallskip
    \[
    \hspace*{-20mm}
    \begin{tikzpicture}[shorten >= -2pt, shorten <= -2pt,scale=1.5]
      \tikzstyle{every node}=[font=\small]
      \tikzstyle{f} = [faded]
      \begin{scope}[shift={(0,0)}]
        \node (Q16) at (0,4) {$G$};
        \node (r2-s) at (-1.25,3) {$\<\rho^2,\sigma\>$};
        \node (r) at (0,3) {$\<\rho\>$};
        \node (r2-rs) at (1.25,3) {$\<\rho^2,\rho\sigma\>$};
        \node (s) at (-2.5,2) {$\<\rho^4,\sigma\>$};
        \node (r2s) at (-1.25,2) {$\<\rho^2\sigma,\rho^4\>$};
        \node (r2) at (0,2) {$\<\rho^2\>$};
        \node (rs) at (1.25,2) {$\<\rho\sigma\>$};
        \node (r3s) at (2.5,2) {$\<\rho^3\sigma\>$};
        \node (r4) at (0,1) {$\<\rho^4\>$};
        \node (a) at (-3,1) {$\<\sigma\>$};
        \node (b) at (-2.25,1) {$\<\rho^4\sigma\>$};
        \node (c) at (-1.5,1) {$\<\rho^2\sigma\>$};
        \node (d) at (-.75,1) {$\<\rho^6\sigma\>$};
        \node (1) at (0,0) {$\<1\>$};
        \draw[f] (Q16) to (r2-s); \draw[f] (Q16) to (r); \draw[f] (Q16) to (r2-rs);
        \draw[f] (r2-s) to (s); \draw[f] (r2-s) to (r2s); \draw[f] (r2-s) to (r2);
        \draw[f] (r) to (r2);
        \draw[f] (r2-rs) to (r2); \draw[f] (r2-rs) to (rs); \draw[f] (r2-rs) to (r3s);
        \draw[f] (r4) to (s); \draw[f] (r4) to (r2s); \draw[f] (r4) to (r2);
        \draw[f] (r4) to (rs); \draw[f] (r4) to (r3s); \draw[f] (1) to (r4);
        \draw[f] (a) to (1); \draw[f] (b) to (1); \draw[f] (c) to (1); \draw[f] (d) to (1);
        \draw[f] (s) to (a); \draw[f] (s) to (b); \draw[f] (r2s) to (c);
        \draw[f] (r2s) to (d);
      \end{scope}
      %
      \begin{scope}[shift={(6,0)},shorten >= 2pt, shorten <= 2pt]
        \node (Q16) at (0,4) {$\Q$};
        \node[white] (r2-s) at (-1.25,3) {$\<r^2,s\>$};
        \node[white] (r) at (0,3) {$\<r\>$};
        \node[white] (r2-rs) at (1.25,3) {$\<r^2,rs\>$};
        \node[white] (s) at (-2.5,2) {$\<r^4,s\>$};
        \node[white] (r2s) at (-1.25,2) {$\<r^2s,r^4\>$};
        \node[white] (r2) at (0,2) {$\<r^2\>$};
        \node[white] (rs) at (1.25,2) {$\<rs\>$};
        \node[white] (r3s) at (2.5,2) {$\<r^3s\>$};
        \node[white] (r4) at (0,1) {$\<r^4\>$};
        \node[white] (a) at (-3,1) {$\<s\>$};
        \node[white] (b) at (-2.25,1) {$\<r^4s\>$};
        \node[white] (c) at (-1.5,1) {$\<r^2s\>$};
        \node[white] (d) at (-.75,1) {$\<r^6s\>$};
        \node (1) at (0,0) {$\Q(\sqrt[8]{2},\zeta)$};
        \draw[f] (Q16) to (r2-s); \draw[f] (Q16) to (r); \draw[f] (Q16) to (r2-rs);
        \draw[f] (r2-s) to (s); \draw[f] (r2-s) to (r2s); \draw[f] (r2-s) to (r2);
        \draw[f] (r) to (r2);
        \draw[f] (r2-rs) to (r2); \draw[f] (r2-rs) to (rs); \draw[f] (r2-rs) to (r3s);
        \draw[f] (r4) to (s); \draw[f] (r4) to (r2s); \draw[f] (r4) to (r2);
        \draw[f] (r4) to (rs); \draw[f] (r4) to (r3s); \draw[f] (1) to (r4);
        \draw[f] (a) to (1); \draw[f] (b) to (1); \draw[f] (c) to (1); \draw[f] (d) to (1);
        \draw[f] (s) to (a); \draw[f] (s) to (b); \draw[f] (r2s) to (c);
        \draw[f] (r2s) to (d);
      \end{scope}
    \end{tikzpicture}
    \]   
    It is helpful to know that the proper subfields of
    $\Q(\sqrt[8]{2},i)$ are: {\small $\Q$, $\Q(i)$, $\Q(\sqrt{2})$,
      $\Q(\sqrt[4]{2})$, $\Q(\sqrt[8]{2})$, $\Q(\sqrt{2}i)$,
      $\Q(\sqrt[4]{2}i)$, $\Q(\sqrt[8]{2}i)$, $\Q(\sqrt{2},i)$,
      $\Q(\sqrt[4]{2},i)$, $\Q((1+i)\sqrt[4]{2})$,
      $\Q((1-i)\sqrt[4]{2})$, $\Q(\zeta\sqrt[8]{2})$,
      $\Q(\zeta^3\sqrt[8]{2})$}.
    
  \item Circle each subfield $E$ that is a normal extension of $\Q$,
    and find a polynomial whose splitting field over $\Q$ is $E$.
  \end{enumerate}

\newpage
  

\item The Galois group of $f(x)=x^5-2$ is generated by the
  automorphisms
  \[
  \left\{\begin{array}{ll}
  \rho\colon\sqrt[5]{2}\longmapsto\zeta\sqrt[5]{2} \\
  \rho\colon\;\;\;\zeta\longmapsto\zeta
  \end{array}\right.\qquad\qquad
  \left\{\begin{array}{ll}
  \sigma\colon\sqrt[5]{2}\longmapsto\sqrt[5]{2} \\
  \sigma\colon\;\;\;\, \zeta\longmapsto\zeta^2\,,
  \end{array}\right.
  \]
  and is isomorphic to the affine general linear group
  \[
  \AGL_2(\Z_5)=\<\rho,\sigma\mid\rho^5=1,\;\sigma^4=1,
  \;\rho\sigma=\sigma\rho^3\>.
  \]
  Two Cayley graphs are shown below.
  \[
  \hspace*{-10mm}
  \begin{tikzpicture}[scale=1.5,auto]
    \tikzstyle{v-small} = [circle, draw, fill=lightgrey,inner sep=0pt,
      minimum size=6mm]
    \tikzstyle{every node}=[font=\tiny]
    %% First semi-direct product, rewired edges
    \begin{scope}[shift={(0,0)}]
      \node (1) at (0,3) [v-small] {$1$};
      \node (a) at (1,3) [v-small] {$\rho$};
      \node (a2) at (2,3) [v-small] {$\rho^2$};
      \node (a3) at (3,3) [v-small] {$\rho^3$};
      \node (a4) at (4,3) [v-small] {$\rho^4$};
      \node (b) at (0,2) [v-small] {$\sigma$};
      \node (ab) at (1,2) [v-small] {$\rho\sigma$};
      \node (a2b) at (2,2) [v-small] {$\rho^2\sigma$};
      \node (a3b) at (3,2) [v-small] {$\rho^3\sigma$};
      \node (a4b) at (4,2) [v-small] {$\rho^4\!\sigma$};
      \node (b2) at (0,1) [v-small] {$\sigma^2$};
      \node (ab2) at (1,1) [v-small] {$\rho\sigma^2$};
      \node (a2b2) at (2,1) [v-small] {$\rho^2\!\sigma^2$};
      \node (a3b2) at (3,1) [v-small] {$\rho^3\!\sigma^2$};
      \node (a4b2) at (4,1) [v-small] {$\rho^4\!\sigma^2$};
      \node (b3) at (0,0) [v-small] {$\sigma^3$};
      \node (ab3) at (1,0) [v-small] {$\rho\sigma^3$};
      \node (a2b3) at (2,0) [v-small] {$\rho^2\!\sigma^3$};
      \node (a3b3) at (3,0) [v-small] {$\rho^3\!\sigma^3$};
      \node (a4b3) at (4,0) [v-small] {$\rho^4\!\sigma^3$};
      %\node at (0,0) {\small $\varphi^3$};
      \draw [b] (1) to (b);
      \draw [b] (b) to (b2);
      \draw [b] (b2) to (b3);
      \draw [b] (b3) to[bend left] (1);
      %%
      \draw [b] (a) to (ab);
      \draw [b] (ab) to (ab2);
      \draw [b] (ab2) to (ab3);
      \draw [b] (ab3) to [bend left] (a);
      %%
      \draw [b] (a2) to (a2b);
      \draw [b] (a2b) to (a2b2);
      \draw [b] (a2b2) to (a2b3);
      \draw [b] (a2b3) to [bend left] (a2);
      %%
      \draw [b] (a3) to (a3b);
      \draw [b] (a3b) to (a3b2);
      \draw [b] (a3b2) to (a3b3);
      \draw [b] (a3b3) to [bend right] (a3);
      %%
      \draw [b] (a4) to (a4b);
      \draw [b] (a4b) to (a4b2);
      \draw [b] (a4b2) to (a4b3);
      \draw [b] (a4b3) to [bend right] (a4);
      %%
      \draw [r] (1) to (a);
      \draw [r] (a) to (a2);
      \draw [r] (a2) to (a3);
      \draw [r] (a3) to (a4);
      \draw [r] (a4) to[bend right=25] (1);
      %%
      \draw [r] (b) to[bend left] (a2b);
      \draw [r] (a2b) to[bend left] (a4b);
      \draw [r] (a4b) to[bend left] (ab);
      \draw [r] (ab) to[bend left] (a3b);
      \draw [r] (a3b) to[bend left] (b);
      %%
      \draw [r] (b2) to[bend right=25] (a4b2);
      \draw [r] (a4b2) to (a3b2);
      \draw [r] (a3b2) to (a2b2);
      \draw [r] (a2b2) to (ab2);
      \draw [r] (ab2) to (b2);
      %%
      \draw [r] (b3) to[bend right] (a3b3);
      \draw [r] (a3b3) to[bend right] (ab3);
      \draw [r] (ab3) to[bend right] (a4b3);
      \draw [r] (a4b3) to[bend right] (a2b3);
      \draw [r] (a2b3) to [bend right] (b3);
    \end{scope}
    %%
    \begin{scope}[shift={(6,0)}]
      %% First semi-direct product, labeled nodes edges
      \node (1) at (0,3) [v-small] {$1$};
      \node (a) at (1,3) [v-small] {$\rho$};
      \node (a2) at (2,3) [v-small] {$\rho^2$};
      \node (a3) at (3,3) [v-small] {$\rho^3$};
      \node (a4) at (4,3) [v-small] {$\rho^4$};
      \node (b) at (0,2) [v-small] {$\sigma$};
      \node (ab) at (3,2) [v-small] {$\rho\sigma$};
      \node (a2b) at (1,2) [v-small] {$\rho^2\!\sigma$};
      \node (a3b) at (4,2) [v-small] {$\rho^3\!\sigma$};
      \node (a4b) at (2,2) [v-small] {$\rho^4\!\sigma$};
      \node (b2) at (0,1) [v-small] {$\sigma^2$};
      \node (ab2) at (4,1) [v-small] {$\rho\sigma^2$};
      \node (a2b2) at (3,1) [v-small] {$\rho^2\!\sigma^2$};
      \node (a3b2) at (2,1) [v-small] {$\rho^3\!\sigma^2$};
      \node (a4b2) at (1,1) [v-small] {$\rho^4\!\sigma^2$};
      \node (b3) at (0,0) [v-small] {$\sigma^3$};
      \node (ab3) at (2,0) [v-small] {$\rho\sigma^3$};
      \node (a2b3) at (4,0) [v-small] {$\rho^2\!\sigma^3$};
      \node (a3b3) at (1,0) [v-small] {$\rho^3\!\sigma^3$};
      \node (a4b3) at (3,0) [v-small] {$\rho^4\!\sigma^3$};
      %\node at (0,0) {\small $\varphi^3$};
      \draw [b] (1) to (b);
      \draw [b] (b) to (b2);
      \draw [b] (b2) to (b3);
      \draw [b] (b3) to [bend left] (1);
      %%
      \draw [b] (a) to (ab);
      \draw [b] (ab) to (ab2);
      \draw [b] (ab2) to (ab3);
      \draw [b] (ab3) to (a);
      %%
      \draw [b] (a2) to (a2b);
      \draw [b] (a2b) to (a2b2);
      \draw [b] (a2b2) to (a2b3);
      \draw [b] (a2b3) to (a2);
      %%
      %%
      \draw [b] (a4) to (a4b);
      \draw [b] (a4b) to (a4b2);
      \draw [b] (a4b2) to (a4b3);
      \draw [b] (a4b3) to (a4);
      %%
      \draw [r] (1) to (a); \draw [r] (a) to (a2); \draw [r] (a2) to (a3);
      \draw [r] (a3) to (a4);
      \draw [r] (a4) to[bend right=25] (1);
      %%
      \draw [r] (b) to (a2b);
      \draw [r] (a2b) to (a4b);
      \draw [r] (a4b) to (ab);
      \draw [r] (ab) to (a3b);
      \draw [r] (a3b) to[bend right=25] (b);
      %%
      \draw [r] (b2) to (a4b2);
      \draw [r] (a4b2) to (a3b2);
      \draw [r] (a3b2) to (a2b2);
      \draw [r] (a2b2) to (ab2);
      \draw [r] (ab2) to[bend left=25] (b2);
      %%
      %%
      \draw [r] (b3) to (a3b3);
      \draw [r] (a3b3) to (ab3);
      \draw [r] (ab3) to (a4b3);
      \draw [r] (a4b3) to (a2b3);
      \draw [r] (a2b3) to[bend left=25] (b3);
    \end{scope}
  \end{tikzpicture}
  \]
  
  \begin{enumerate}
  \item Draw the action graph of $G=\Gal(x^5-2)$ acting on the set $S$
    of roots of $x^5-2$, like what was done in the previous problem.
  \item For each subgroup $H\leq\Gal(x^5-2)$, find the largest
    subgroup of $\Q(\sqrt[5]{2},\zeta)$ fixed by $H$, and write it in the
    corresponding place on the subfield lattice on the
    right.
    \[
    \hspace*{-8mm}
    \begin{tikzpicture}[shorten >= -0pt, shorten <= -0pt,scale=1.75,yscale=1.2]
      \tikzstyle{every node}=[font=\normalsize]
      \tikzstyle{f} = [faded]
      \begin{scope}[shift={(0,0)}]
        \node (Fr20) at (0,4) {$\Gal(x^5-2)$};
        \node(s2-t) at (-.15,2.9) {$\<\rho,\!\sigma^2\>$};
        \node (t) at (-1.4,2.3) {$\<\rho\>$};
        \node (s) at (-.7,2) {$\<\sigma\>$};
        \node (st) at (.25,2) {$\<\rho^2\sigma\>$};
      \node (s3t) at (1.15,2) {$\<\rho^4\sigma\>$};
      \node (ts) at (1.8,2) {$\<\rho\sigma\>$};
      \node (ts3) at (2.5,2) {$\<\rho^3\sigma\>$};
      \node(s2) at (-.45,1.1) {$\<\sigma^2\>$};
      \node(ts2) at (.25,1.1) {$\<\rho\sigma^2\>$};
      \node(sts) at (1.15,1.1) {$\<\rho^2\sigma^2\>$};
      \node(s2t2) at (1.8,1.1) {$\<\rho^3\sigma^2\>$};
      \node(s2t) at (2.5,1.1) {$\<\rho^4\sigma^2\>$};
      \node (1) at (0,0) {$\<1\>$};
      \draw[bend right=5,f] (Fr20) to (s);
      \draw[f] (Fr20) to (st);
      \draw[f] (Fr20) to (s3t);
      \draw[f] (Fr20) to (ts);
      \draw[f] (Fr20) to (ts3);
      \draw[f] (Fr20) to (s2-t);
      %%
      \draw[f](s2-t) to (t); \draw[f](t) to (1);
      \draw[f](s2-t) to (s2); \draw[f](s2-t) to (ts2);
      \draw[f](s2-t) to (sts);
      \draw[f](s2-t) to (s2t2);
      \draw[bend left=10,f] (s2-t) to (s2t);
      %%
      \draw[faded] (s) to (s2); \draw[faded] (st) to (ts2);
      \draw[faded] (s3t) to (sts); \draw[faded] (ts) to (s2t2);
      \draw[faded] (ts3) to (s2t);
      %%
      \draw[f](1) to (s2); \draw[f](1) to (ts2); \draw[f](1) to (sts);
      \draw[f](1) to (s2t2); \draw[f](1) to (s2t);
    \end{scope}
    %
    \begin{scope}[shift={(5,0)},shorten >= 2pt, shorten <= 2pt]
      \node (Fr20) at (0,4) {$\Q$};
      \node[white](s2-t) at (-.15,2.9) {$\<s^2,\!t\>$};
      \node[white] (t) at (-1.4,2.3) {$\<t\>$};
      \node[white] (s) at (-.7,2) {$\<s\>$};
      \node[white] (st) at (.25,2) {$\<st\>$};
      \node[white] (s3t) at (1.15,2) {$\<s^3t\>$};
      \node[white] (ts) at (1.8,2) {$\<ts\>$};
      \node[white] (ts3) at (2.5,2) {$\<ts^3\>$};
      \node[white](s2) at (-.45,1.1) {$\<s^2\>$};
      \node[white](ts2) at (.25,1.1) {$\<ts^2\>$};
      \node[white](sts) at (1.15,1.1) {$\<sts\>$};
      \node[white](s2t2) at (1.8,1.1) {$\<s^2t^2\>$};
      \node[white](s2t) at (2.5,1.1) {$\<s^2t\>$};
      \node (1) at (0,0) {$\Q(\sqrt[5]{2},\zeta)$};
      \draw[bend right=5,f] (Fr20) to (s);
      \draw[f] (Fr20) to (st);
      \draw[f] (Fr20) to (s3t);
      \draw[f] (Fr20) to (ts);
      \draw[f] (Fr20) to (ts3);
      \draw[f] (Fr20) to (s2-t);
      %%
      \draw[f](s2-t) to (t); \draw[f](t) to (1);
      \draw[f](s2-t) to (s2); \draw[f](s2-t) to (ts2);
      \draw[f](s2-t) to (sts);
      \draw[f](s2-t) to (s2t2);
      \draw[bend left=10,f] (s2-t) to (s2t);
      %%
      \draw[faded] (s) to (s2); \draw[faded] (st) to (ts2);
      \draw[faded] (s3t) to (sts); \draw[faded] (ts) to (s2t2);
      \draw[faded] (ts3) to (s2t);
      %%
      \draw[f](1) to (s2); \draw[f](1) to (ts2); \draw[f](1) to (sts);
      \draw[f](1) to (s2t2); \draw[f](1) to (s2t);
    \end{scope}
    \end{tikzpicture}
    \]
  \end{enumerate}

  \newpage

  \item Repeat the previous problem for the Galois group of
    $f(x)=x^6-2$, which is generated by the automorphisms
  \[
  \left\{\begin{array}{ll}
  \rho\colon\sqrt[6]{2}\longmapsto\zeta\sqrt[6]{2} \\
  \rho\colon\;\;\;\zeta\longmapsto\zeta
  \end{array}\right.\qquad\qquad
  \left\{\begin{array}{ll}
  \sigma\colon\sqrt[6]{2}\longmapsto\sqrt[6]{2} \\
  \sigma\colon\;\;\;\, \zeta\longmapsto\overline\zeta\,,
  \end{array}\right.
  \]
  and is isomorphic to the dihedral group
  \[
  D_6=\<\rho,\sigma\mid\rho^6=1,\;\sigma^2=1,
  \;\rho\sigma=\sigma\rho^5\>.
  \]
  The subgroup and blank subfield lattice are shown below.
  %%
  \newcommand\Hf{6.495} % height of D_6
  \newcommand\He{5.196} % height of C_6
  \newcommand\Hd{3.897} % height of V_4
  \newcommand\Hc{2.598} % height of C_3
  \newcommand\Hb{1.299} % height of C_2
  \newcommand\Ha{0} % height of C_1
  \[
  \hspace*{-8mm}
  %%
  %% ------------- Subgroup lattice of D_6 ------------------------
  %%
  \begin{tikzpicture}
  \begin{scope}[shift={(0,0)},scale=1.5,xscale=.95]
    \tikzstyle{every node}=[font=\small]
    \node(D6) at (.75,\Hf) {$D_6$};
    \node(r2-f) at (-.25,\He) {$\<\rho^2,\sigma\>$};
    \node(r2-rf) at (.75,\He) {$\<\rho^2,\rho\sigma\>$};
    \node(r) at (-1.25,\He) {$\<\rho\>$};
    \node(r2) at (-1.25,\Hc) {$\<\rho^2\>$};
    %%   
    \node(r3-f) at (2.4,\Hd) {\scriptsize $\<\rho^3\!,\!\sigma\>$};
    \node(r3-rf) at (3.1,\Hd) {\scriptsize $\<\rho^3\!,\rho\sigma\>$};
    \node(r3-r2f) at (3.8,\Hd) {\scriptsize $\<\rho^3\!,\!\rho^2\!\sigma\>$};
     \node(r3) at (0,\Hb) {$\<\rho^3\>$};
    \node(f) at (.6,\Hb) {$\<\sigma\>$};
    \node(r4f) at (1.2,\Hb) {$\<\rho^4\!\sigma\>$};
    \node(r2f) at (1.8,\Hb) {$\<\rho^2\!\sigma\>$};
    \node(r3f) at (2.4,\Hb) {$\<\rho^3\!\sigma\>$};
    \node(rf) at (3.1,\Hb) {$\<\rho\sigma\>$};
    \node(r5f) at (3.8,\Hb) {$\<\rho^5\!\sigma\>$};
    \node (1) at (0,\Ha) {$\<1\>$};
    \draw[faded] (D6) -- (r2-f);
    \draw[faded] (D6) -- (r);
    \draw[faded] (D6) to (r2-rf);
    \draw[faded] (D6) to (r3-f);
    \draw[faded] (D6) -- (r3-rf);
    \draw[faded] (D6) to (r3-r2f);
    \draw[faded] (r2-f) to (f);
    \draw[faded] (r2-f) -- (r2f);
    \draw[faded] (r2-f) to (r4f);
    \draw[faded] (r2-f) to (r2);
    \draw[faded] (r) -- (r2);
    \draw[faded] (r) -- (r3);
    \draw[faded] (r2-rf) -- (r2);
    \draw[faded] (r2-rf) to (rf);
    \draw[faded] (r2-rf) to (r3f);
    \draw[faded] (r2-rf) to (r5f);
    \draw[faded] (r3-f) -- (f);
    \draw[faded] (r3-f) to (r3);
    \draw[faded] (r3-f) to (r3f);
    \draw[faded] (r3-rf) to (rf);
    \draw[faded] (r3-rf) to (r3);
    \draw[faded] (r3-rf) to (r4f);
    \draw[faded] (r3-r2f) to (r2f);
    \draw[faded] (r3-r2f) to (r3);
    \draw[faded] (r3-r2f) to (r5f);
    \draw[faded] (r2) -- (1);
    \draw[faded] (f) -- (1);
    \draw[faded] (rf) to (1);
    \draw[faded] (r2f) to (1);
    \draw[faded] (r3f) to (1);
    \draw[faded] (r4f) to (1);
    \draw[faded] (r5f) to (1);
    \draw[faded] (r3) -- (1);
  \end{scope}
  %%
  \begin{scope}[shift={(9,0)},scale=1.5,xscale=.95]
    \tikzstyle{every node}=[font=\small]
    \node(D6) at (.75,\Hf) {$\Q$};
    \node[white](r2-f) at (-.25,\He) {$\<\rho^2,\sigma\>$};
    \node[white](r2-rf) at (.75,\He) {$\<\rho^2,\rho\sigma\>$};
    \node[white](r) at (-1.25,\He) {$\<\rho\>$};
    \node[white](r2) at (-1.25,\Hc) {$\<\rho^2\>$};
    %%   
    \node[white](r3-f) at (2.4,\Hd) {\scriptsize $\<\rho^3\!,\!\sigma\>$};
    \node[white](r3-rf) at (3.1,\Hd) {\scriptsize $\<\rho^3\!,\rho\sigma\>$};
    \node[white](r3-r2f) at (3.8,\Hd) {\scriptsize $\<\rho^3\!,\!\rho^2\!\sigma\>$};
     \node[white](r3) at (0,\Hb) {$\<\rho^3\>$};
    \node[white](f) at (.6,\Hb) {$\<\sigma\>$};
    \node[white](r4f) at (1.2,\Hb) {$\<\rho^4\!\sigma\>$};
    \node[white](r2f) at (1.8,\Hb) {$\<\rho^2\!\sigma\>$};
    \node[white](r3f) at (2.4,\Hb) {$\<\rho^3\!\sigma\>$};
    \node[white](rf) at (3.1,\Hb) {$\<\rho\sigma\>$};
    \node[white](r5f) at (3.8,\Hb) {$\<\rho^5\!\sigma\>$};
    \node (1) at (0,\Ha) {$\Q(\sqrt[6]{2},\zeta)$};
    \draw[faded] (D6) -- (r2-f);
    \draw[faded] (D6) -- (r);
    \draw[faded] (D6) to (r2-rf);
    \draw[faded] (D6) to (r3-f);
    \draw[faded] (D6) -- (r3-rf);
    \draw[faded] (D6) to (r3-r2f);
    \draw[faded] (r2-f) to (f);
    \draw[faded] (r2-f) -- (r2f);
    \draw[faded] (r2-f) to (r4f);
    \draw[faded] (r2-f) to (r2);
    \draw[faded] (r) -- (r2);
    \draw[faded] (r) -- (r3);
    \draw[faded] (r2-rf) -- (r2);
    \draw[faded] (r2-rf) to (rf);
    \draw[faded] (r2-rf) to (r3f);
    \draw[faded] (r2-rf) to (r5f);
    \draw[faded] (r3-f) -- (f);
    \draw[faded] (r3-f) to (r3);
    \draw[faded] (r3-f) to (r3f);
    \draw[faded] (r3-rf) to (rf);
    \draw[faded] (r3-rf) to (r3);
    \draw[faded] (r3-rf) to (r4f);
    \draw[faded] (r3-r2f) to (r2f);
    \draw[faded] (r3-r2f) to (r3);
    \draw[faded] (r3-r2f) to (r5f);
    \draw[faded] (r2) -- (1);
    \draw[faded] (f) -- (1);
    \draw[faded] (rf) to (1);
    \draw[faded] (r2f) to (1);
    \draw[faded] (r3f) to (1);
    \draw[faded] (r4f) to (1);
    \draw[faded] (r5f) to (1);
    \draw[faded] (r3) -- (1);
  \end{scope}
\end{tikzpicture}
\]

  
\end{enumerate}

\end{document}
