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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due November 11 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2019 $|$ M.~Macauley}}



\begin{document}

\noindent Read Chapters 10.1--10.5 of \emph{Visual Group Theory} or Chapters 20.1--20.2 of \emph{AATA}. Then write up solutions to the following exercises. \bigskip


\begin{enumerate}

\item There is a unique \emph{finite} 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. \vspace{-2mm}
  
  \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
    $\F_7$. \bigskip
  \end{enumerate}



  \item The field $\Q(\sqrt[4]{3},i)$ is called the \emph{splitting
    field} of the polynomial $f(x)=x^4-3$ over $\Q$ because it is the
    smallest extension field of $\Q$ that contains all roots of $f(x)$.
    \begin{enumerate}
    \item Sketch the roots of $f(x)=x^4-3$ in the complex plane. Write
      each one as $a+bi$, where $a,b\in\R$. Additionally, write each
      root in polar form: $z=Re^{i\theta}$.
    \item Find a basis for the extension field $\Q(\sqrt[4]{3})$ of
      $\Q$ and compute its dimension as a $\Q$-vector space. 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. 
    \item Is $\Q(\sqrt[4]{3})$ the splitting field of some polynomial
      $g(x)$ over $\Q$? If yes, find $g(x)$. If no, explain why not.
    \item Find a basis for
      $\Q(\sqrt[4]{3},i):=\Q(\sqrt[4]{3})(i)=\Q(i)(\sqrt[4]{3})$ over
      each of the fields $\Q(\sqrt[4]{3})$, $\Q(i)$, and $\Q$. What is
      its dimension as a vector space over each of these fields?
    \item $\Q(\sqrt[4]{3},i)$ is the splitting field of what
      polynomial over $\Q(\sqrt[4]{3})$? And of what polynomial over
      $\Q(i)$? \bigskip
    \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. For each edge in this
  lattice, which corresponds to an extension field $E\supseteq F$,
  write the degree of the extension of $E$ over $F$, which by
  definition is the dimension of $E$ as an $F$-vector-space.



\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 $\alpha,\beta\in\Q(\sqrt{2})$:
  \[
  \phi(\alpha+\beta)=\phi(\alpha)+\phi(\beta)\,,\qquad
  \phi(\alpha\cdot\beta)=\phi(\alpha)\cdot\phi(\beta)\,.
  \]

  \vspace{0mm}

  
\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
      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}

