% Final for Math 362, given on May 4, 1998 from 4:15--6:15pm by
% Joachim Rosenthal.
\documentclass[12pt]{letter}
\usepackage{amssymb}
\textwidth 6.8in \textheight 9.4in
\topmargin -0.95in \oddsidemargin -0.2in
\newcommand{\R}{{\mathbb R}} \newcommand{\F}{{\mathbb F}}
\newcommand{\Z}{{\mathbb Z}} \newcommand{\Q}{{\mathbb Q}}
\newcommand{\K}{{\mathbb K}} \newcommand{\C}{{\mathbb C}}
\newcommand{\vier}[4]{\left[ \begin{array}{ccc} #1 &\;& #2 \\ #3
&\;& #4 \end{array} \right]}
\newcommand{\newpageSp}{\newpage}
%\newcommand{\newpageSp}{} %% For one page version
\begin{document}
\normalsize
\baselineskip 0.47cm %\thispagestyle{empty}
\begin{center}
{\bf Final Exam for Math 362, Honors Algebra IV}\\
May 4, 1998, 4:15--6:15pm.
\end{center}
\begin{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (10 pts) Write down an irreducible polynomial which is not
normal. Explain why your example has the stated properties.
\vfill
(10 pts) Is $[\C :\R]$ a Galois extension? Give a detailed
answer.
\vfill
\newpageSp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (10 pts) Let
$$
H:= {\rm rowsp}_{\Z} \left(\begin{array}{ccc}
2 & 0 & 4\\
0 & 2 & 0 \\
4 & 0 & 10
\end{array}\right).
$$
Compute the invariant factor decomposition of the abelian group
${\Z}^3/H$.
\vfill
(10 pts) Count the number of non-isomorphic abelian groups of
order 720.
\vfill
\newpageSp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (10 pts) Show that $x^3-t\in\Z_3(t)[x]$ is an irreducible
and inseparable polynomial.
\vfill
(10 pts) Write down a nontrivial field automorphism
$GF(1024)\longrightarrow GF(1024)$. \vfill
\newpageSp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (10 pts) Find the number of all $[5,3]$ linear codes
defined over $\Z_2$. Alternatively count the number of
subspaces $V\subset \left(\Z_2\right)^5$ with $\dim V=3$.
\vfill
(10 pts) Give an example of a group of order 60 which is not
solvable (=soluble).
\vfill
\newpageSp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (20 pts) Prove in detail that $GF(2^{30})\supset GF(2)$ is
a simple extension.
\newpageSp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (20 pts) Prove that $\Q(\sqrt{5}+\sqrt{7})\supset\Q$ is a
Galois extension. Then compute the Galois group
$\Gamma[\Q(\sqrt{5}+\sqrt{7}):\Q ]$ and work out the Galois
correspondence indicating on one side the subgroups of $\Gamma$
and on the other side the subfields of $\Q(\sqrt{5}+\sqrt{7})$.
\newpageSp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item a) (10 pts) Describe in detail the splitting field
$\F\subset\C$ of the polynomial $x^8-1\in\Q[x]$.
\vfill
b) (10pts) Compute the Galois group $\Gamma[\F:\Q]$. What is
the degree $[\F:\Q]$?
\vfill \newpageSp
c) (10pts) Work out the Galois correspondence indicating on one
side the subgroups of $\Gamma[\F:\Q]$ and on the other side the
subfields of $\F$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{enumerate}
\end{document}