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



\begin{document}

$\;$ %\vspace{-5mm}

\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 Recall that a group $G$ is called \emph{simple} if its only
  normal subgroups are $G$ and $\{e\}$.
  \begin{enumerate}
  \item Show that there is no simple group of order $45=3^2\cdot
    5$. 
  \item Show that there is no simple group of order $pq$, where
    $p<q$ and are both prime.
  \item Show that there is no simple group of order $12=2^2\cdot 3$.
  \item Show that there is no simple group of order $56=2^3\cdot
    7$. 
  \item Show that there is no simple group of order $108=2^2\cdot
    3^3$. \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
      the dimension of $\Q(\sqrt[4]{3},i)$ 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. \bigskip



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


\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}

