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\pagestyle{fancy} \lhead{{\sf Homework 12 $|$ Due Tuesday, November 21
}} \rhead{\thepage} \cfoot{{\sf Math 4120/6120 $|$ Modern Algebra $|$
    Fall 2017 $|$ M.~Macauley}}



\begin{document}

$\;$ \vspace{-3mm}

\noindent Read the following, which can all be found either in the
textbook or on the course website.
\begin{itemize}
\item Chapters 10.6--10.7 of \emph{Visual Group Theory}
  (VGT). \vspace{-2mm}
\item VGT Exercises 10.3, 10.7, 10.16--10.18, 10.20,
  10.22--10.28. \smallskip

\end{itemize} 


\noindent Write up solutions to the following exercises.

\begin{enumerate}


  \item Recall that the splitting field of $f(x)=x^4-3$ is
    $K=\Q(\sqrt[4]{3},i)$. Since this is a degree-$8$ extension over
    $\Q$ (see HW 11), its Galois group has order $8$.
    \begin{enumerate}[(i)]
    \item Sketch the roots of $f(x)$ in the complex plane. 
    \item Compute the Galois group of $f(x)$. Write down two
      automorphisms, $r$ and $f$, that generate it. It suffices to
      specify where they send the generators $\sqrt[4]{3}$ and $i$.
    \item Draw the subgroup lattice of $G$.  Each subgroup should be
      expressed by its generators, rather than what subgroup it is
      isomorphic to. Label the edges by index,
      and circle the subgroups that are normal in $G$.
    \item Draw the subfield lattice of $K$. Label the edges by degree,
      and circle the subfields that are normal extensions of
      $\Q$. [\emph{Hint}: The two subfields that are ``easiest'' to
        overlook are $\Q((1+i)\sqrt[4]{3})$ and $\Q((1-i)\sqrt[4]{3})$.]
    \item For each intermediate subfield $\Q\subseteq F\subseteq K$,
      write down the largest subgroup of $G$ that fixes $F$.
    \item For each subgroup $H\leq G$, write down the largest
      intermediate subfield fixed by $H$.
    \item For each normal extension $F$ of $\Q$, find a polynomial
      $f(x)$ whose splitting field is $F$.
    \item For each non-normal extension $E$ of $\Q$, find a polynomial
      that has one, but not all, of its roots in $E$. \bigskip
    \end{enumerate}



  \item The roots of $f(x)=x^n-1$ are the $n$ complex
    numbers $C_n:=\{e^{2k\pi i/n}:k=0,1,\dots,n-1\}$, and are called the
    \emph{$n^{\rm th}$ roots of unity}. A \emph{primitive} root of
    unity is $\zeta=e^{2k\pi i/n}$ for which $\gcd(n,k)=1$. It is easy
    to see that $\Q(\zeta)$ is the splitting field of $x^n-1$.
    \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
      $n$. Next to each root, write its order, as an element of $C_n$.
      Make it clear (e.g., star, or draw darker) which are
      the primitive roots of unity.
    \item Prove that for any primitive root of unity $\zeta_k=e^{2k\pi
      i/n}$, the mapping
      \[
      \phi_k\colon\Q(\zeta)\to\Q(\zeta)\,,\qquad
      \phi_k(\zeta)=\zeta^k
      \]
      is a field automorphism. That is, prove that $\phi_k$ is a
      surjective field homomorphism (every nonzero field homomorphism
      is injective). Why is this not an automorphism if $\gcd(n,k)\neq
      1$?
    \item Make a muliplication table of $\Gal(x^n-1)$ for
      $n=3,\dots,8$.
    \item Describe the group $\Gal(x^n-1)$. This is a class of
      groups that we have previously encountered.
    \end{enumerate}
    


  \item For each of the following polynomials, determine if it is
    irreducible. If it is not, then factor it into irreducible factors.
    \begin{enumerate}
    \item $f(x)=x^4-10x^3+12x^2-8x+6$ over $\Q$.
    \item $f(x)=x^4+x^3+x^2+x+1$ [\emph{Hint}: Let $u=x+1$, and change
      variables.]
    \item $f(x)=x^5-1$ over $\Q$. 
    \item $f(x)=x^6-1$ over $\Q$. [\emph{Hint}: Google ``cyclotomic
      polynomial.'']
    \item $f(x)=x^8-1$ over $\Q$.
    \item $f(x)=x^{12}-1$ over $\Q$.
    \item $f(x)=x^3+x^2+x+1$ over $\Z_2$.
    \item $f(x)=x^3+x+2$ over $\Z_3$.
    \end{enumerate}

\end{enumerate}


\end{document}

