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\pagestyle{fancy} \lhead{{\sf Homework 12 $|$ Due April 14 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Spring 2014 $|$ M.~Macauley}}



\begin{document}

$\;$

\noindent Read the following, which can all be found either in the
textbook or on the course website.
\begin{itemize}
\item Chapters 10.1--10.5 of \emph{Visual Group Theory}
  (VGT). \vspace{-2mm}
\item VGT Exercises 10.1, 10.2, 10.4, 10.5, 10.8--10.14, 10.21,
  10.30. \smallskip

\end{itemize} 


\noindent Write up solutions to the following exercises.

\begin{enumerate}


\item There is a unique field $\F_q$ of order $q=p^k$ for every prime
  $p$ and positive integer $k$. For all other $q\in\N$, there is no
  finite field of order $q$. For each of the fields $\F_4$, $\F_5$,
  and $\F_8$, the Cayley diagrams for addition and multiplication are
  shown below, overlayed on the same set of nodes. The solid arrows
  are the Cayley diagrams for addition and the dashed arrows are the
  Cayley diagrams for multiplication.
  
  \tikzstyle{v} = [circle, draw, fill=lightgrey,inner sep=0pt,
    minimum size=5mm] 
  \tikzstyle{g-v} = [circle, draw, fill=darkgreen,inner sep=0pt,
    minimum size=6mm] 
  \tikzstyle{b} = [draw,very thick, Blue]
  \tikzstyle{g} = [draw,very thick, darkgreen]
  \tikzstyle{g-dashed} = [draw,very thick, darkgreen, dashed, -stealth]
  \tikzstyle{r} = [draw, very thick, Red,-stealth,bend left=20]
  \tikzstyle{r2} = [draw, very thick, Red]
  \tikzstyle{p} = [draw, very thick, dashed, darkpurple,-stealth]
  \begin{center}
    \begin{tikzpicture}[scale=1]
      \node (0) at (135:1.5) [v] {{\small $0$}};
      \node (1) at (45:1.5) [v] {\small {$a$}};
      \node (3) at (-135:1.5) [v] {{\small $c$}};
      \node (2) at (-45:1.5) [v] {{\small $b$}};   
      \draw [b] (0) to (1);
      \draw [b] (2) to (3);
      \draw [r2] (0) to (3);
      \draw [r2] (1) to (2);
      \draw [g-dashed,bend left=20] (3) to (1);
      \draw [g-dashed,bend left=40] (1) to (2);
      \draw [g-dashed,bend left=40] (2) to (3);
    \begin{scope}[shift={(5.5,0)}]
      \node (0) at (90:1.5) [v] {{\small $0$}};
      \node (1) at (18:1.5) [v] {\small {$a$}};
      \node (2) at (-54:1.5) [v] {{\small $b$}};
      \node (3) at (234:1.5) [v] {{\small $c$}};
      \node (4) at (162:1.5) [v] {{\small $d$}};
      \draw [r] (0) to (1);
      \draw [r] (1) to (2);
      \draw [r] (2) to (3);
      \draw [r] (3) to (4);
      \draw [r] (4) to (0);
      \draw [g-dashed,bend right=20] (1) to (2);
      \draw [g-dashed,bend right=20] (2) to (4);
      \draw [g-dashed,bend left=20] (4) to (3);
      \draw [g-dashed,bend left=20] (3) to (1);
    \end{scope}
    \begin{scope}[shift={(10,-1.25)}]
      \node (2) at (0,0) [v] {\footnotesize $b$};
      \node (0) at (-0.1,2) [v] {\footnotesize $0$};
      \node (3) at (2,-.4) [v] {\footnotesize $c$};
      \node (1) at (2.1,1.7) [v] {\footnotesize $a$};
      \node (6) at (1,1.1) [v] {\footnotesize $f$};
      \node (4) at (.9,3) [v] {\footnotesize $d$};
      \node (7) at (2.7,.8) [v] {\footnotesize $g$};
      \node (5) at (2.8,2.8) [v] {\footnotesize $e$}; 
      \draw [r2] (0) to (1);
      \draw [r2] (2) to (3);
      \draw [b] (0) to (4);
      \draw [b] (1) to (5);
      \draw [b] (2) to (6);
      \draw [b] (3) to (7);
      \draw [g] (0) to (2);
      \draw [g] (1) to (3);
      \draw [r2] (4) to (5);
      \draw [r2] (6) to (7);
      \draw [g] (4) to (6);
      \draw [g] (5) to (7);
      \draw [p,bend left=20] (1) to (2);
      \draw [p,bend left=10] (2) to (4);
      \draw [p] (4) to (3);
      \draw [p,bend left=35] (3) to (6);
      \draw [p,bend right=40] (6) to (7);
      \draw [p,bend right=40] (7) to (5);
      \draw [p,bend right=55] (5) to (1);
    \end{scope}
    \end{tikzpicture}
  \end{center}
  \begin{enumerate}
  \item For each field above, determine whether or not the addition
    and multiplication operations are in fact, addition and
    multiplication modulo some number. If yes, relabel the vertices
    accordingly. If no, explain why it fails.
  \item Create Cayley diagrams for the finite fields $\F_3$ and
    $\F_7$. \smallskip
  \end{enumerate}



  \item The roots of the polynomial $f(x)=x^n-1$ are the $n$ complex
    numbers $\{e^{2k\pi i/n}:k=0,1,\dots,n-1\}$. These are called the
    \emph{$n^{\rm th}$ roots of unity}.
    \begin{enumerate}
      \item For each $n=3,\dots,8$, sketch the $n^{\rm th}$ roots of
        unity in the complex plane. Use a different set of axes for
        each $n$.
      \item The $n^{\rm th}$ roots of unity form a group under
        multiplication. Give a minimal generating set. What group is
        this isomorphic to? \smallskip
    \end{enumerate}
    


  \item 
    \begin{enumerate}
    \item Find a basis for the extension field $\Q(\sqrt[4]{3})$ of
      $\Q$. That is, find a minimal set of
      $v_1,\dots,v_k\in\Q(\sqrt[4]{3})$ such that every
      $x\in\Q(\sqrt[4]{3})$ can be written as a unique linear
      combination of the $v_i$'s. [\emph{Hint}: Start by taking
        $v_1=1$ and $v_2=\sqrt[4]{3}$\dots]
    \item Sketch the roots of the polynomial $f(x)=x^4-3$ in the
      complex plane. Write each one as $a+bi$, where
      $a,b\in\Q$. Additionally, write each root in polar form:
      $z=Re^{i\theta}$.
    \item The field $\Q(\sqrt[4]{3},i)$ is called the \emph{splitting
      field} of $f$ over $\Q$, because it is the smallest extension
      field of $\Q$ that contains the roots of $f$. Find a basis for
      $\Q(\sqrt[4]{3},i)=\Q(\sqrt[4]{3})(i)$ over $\Q$. What is its
      dimension as a $\Q$-vector space?
    \end{enumerate}



\item Thus far in class, we have seen a number of algebraic
  extensions of $\Q$, including:
  \[
  \mbox{
    $\Q(\sqrt{2}),\quad \Q(\sqrt{3}),\quad \Q(\sqrt{6}),\quad
  \Q(\sqrt{2},\sqrt{3}),\quad \Q(i),\quad
  \Q(\sqrt{-3},\sqrt[3]{2}),\quad \Q(\sqrt[4]{3},i),\quad
  \Q(\sqrt[4]{3})$.}
  \]
  Arrange these fields in a subfield lattice, and include $\Q$, $\R$
  and $\C$ as well. Note that there will be (many!) ``missing''
  fields, so only include those listed above. Include the degree of
  each extension, but only for the actual edges in your
  lattice. \smallskip



\item Consider the function 
  \[
  \phi\colon\Q(\sqrt{2})\longto\Q(\sqrt{2})\,,\qquad
  \phi(a+b\sqrt{2})=a-b\sqrt{2}\,.
  \]
  Show that $\phi$ is a field automorphism, meaning that it
  satisfies the following equations for all $\alpha,\beta\in\Q(\sqrt{2})$:
  \[
  \phi(\alpha+\beta)=\phi(\alpha)+\phi(\beta)\,,\qquad
  \phi(\alpha\cdot\beta)=\phi(\alpha)\cdot\phi(\beta)\,.
  \]
  


\item Consider the following extension field of $\Q$:
  \[
  K=\Q(\sqrt{2},i)=\{a+b\sqrt{2}+ci+d\sqrt{2}i\mid a,b,c,d\in\Q\}\,.
  \]
  \begin{enumerate}
    \item Find the Galois group $G=\Gal(K)$ of $K$ over $\Q$. For each
      automorphism $\phi\in G$, write down where it sends the generators
      $\sqrt{2}$ and $i$, and then write down
      \[
      \phi(a+b\sqrt{2}+ci+d\sqrt{2}i)\,.
      \]
    \item Write out a multiplication table for $G$, and a
      minimal generating set.
    \item Write down the subfield lattice of $K$ and the subgroup
      lattice of $G$. Each subgroup should be expressed by its
      generators, rather than what subgroup it is isomorphic to.
    \item For each subgroup $H\leq G$, determine the largest
      subfield of $K$ that $H$ fixes.
    \item For each subfield $F\subseteq K$, determine the largest
      subgroup of $G$ that fixes $F$.
  \end{enumerate}

\end{enumerate}


\end{document}

