%%  answers for 1a, 3.
%% problem and answer for 2d and 2e,
%% approve change in points for proof problems.

\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts}

\input{ExamMacros.tex}

\setlength{\topmargin}{-0.5in}
\setlength{\textheight}{9.5in}
\setlength{\oddsidemargin}{-.2in}
\setlength{\evensidemargin}{-.2in}
\setlength{\textwidth}{6.6in}
\setlength{\parindent}{.2in}

\prof{Peter Cholak and Juan Migliore} \class{Math 222 - Exam 2}
\quizdate{Wednesday, April 25} \pagestyle{empty}

\begin{document}

\heading

\noindent {\bf Except where noted, be sure to show all your work.}

\bigskip

\begin{problem}(4 points each for a total of 20 points) 
  {\bf Define} the following terms: 
  
  a) \emph{irreducible polynomial}. %\vspace{1.5in}

An irreducible polynomial is a polynomial $f$ which cannot be
factored as $f = gh$ where $g$ and $h$ are polynomials and
$0 < \deg (g) < \deg (f)$.

  b) \emph{$GF(p,P(x))$}, where $P(x)$ is irreducible and $p$ is
  prime. %\vspace{1.5in}
  
  Let $v$ be the degree of $P(x)$ and $\alpha$ be a Galois imaginary
  of $P(x)$. Then $GF(p,P(x))$ is the Galois field consisting of all
  the elements of the form
  $a_0+a_1\alpha+a_2\alpha^2+\cdots+a_{v-1}\alpha^{v-1}$ with
  $a_i\in\Bbb{Z}_p$, whose addition and multiplication are defined as
  those for polynomials in one variable modulo the relation
  $P(\alpha)=0$.\par


  c) an \emph{even permutation}.  %\vspace{1.5in}
   
  An even permutation is a permutation which can be written as a
  product of even number of transpositions.
 
  d) a \emph{group}.
%  \vspace{1.5in}
  
  A group is a non-empty set $G$ with a binary operation $\cdot$ on
  its elements satisfying: \par (1) $a\cdot b\in G$ for any $a,b\in
  G$.\par (2) There is $1_G\in G$ such that $a\cdot 1_G=1_G\cdot a=a$
  for $a\in G$.\par (3) $(a\cdot b)\cdot c=a\cdot(b\cdot c)$ for
  $a,b,c\in G$.\par (4) For every $a\in G$, there is $a^{-1}\in G$
  such that $a\cdot a^{-1}=a^{-1}\cdot a=1_G$ ($a^{-1}$ is written as
  $a^{\#}$ by the book) .


  e) The \emph{symmetric group $S_n$}.
  
  The symmetric group $S_n$ is a permutation group consisting of all
  the permutations on the set $\{1,2,\cdots,n\}$, where the binary
operation for the group is composition of permutations.

\end{problem}

%\newpage
  
\begin{problem}(2 points each) Answer {\bf True} or {\bf False} - no work
  required:

 a) The dihedral group $D_4$ is not abelian. %\vspace{1.5in}
  
  True.

\smallskip
  
  b) The order of $(\mathbb{Z}_n, +)$ is $n$. %\vspace{1.5in}
  
  True. $\Bbb{Z}_n=\{0,1,2,\cdots,n-1\}$, containing exactly $n$
  elements.

\smallskip
  
  c) The set of \emph{primitive} $8$th roots of unity in
  $\mathbb{Z}_{17}$ is a group (under multiplication). %\vspace{1.5in}
  
  False. This set contains no identity element. Also, it is not closed
  under the multiplication.

\smallskip
 
  d)  Let $p$ be a prime number.  The multiplicative group $({\mathbb
Z}_p \backslash \{ 0\}, \cdot)$ has at least one element of order
$p$.  (Notice this says $p$ and not $p-1$.) 

False.  Fermat's theorem says that $x^p \equiv_p x$ for all $x \in
{\mathbb Z}_p$, so the only candidate is $x = 1$. But 1 has order 1. 

\smallskip

  e)   Let $p$ be a prime number and let $P(x)$ be a polynomial of
degree $d$ that is irreducible over $\mathbb{Z}_p$.  Let $G = (GF(p,P(x))
\backslash \{0\}, \cdot)$, the multiplicative group of $GF(p,P(x))$. 
Then there must exist an element $x$ of $G$ satisfying $|G| = o(x)$.

True since every Galois field contains a primitive element.




\end{problem}

%\newpage

\begin{problem} (15 points)
  Work over $\mathbb{Z}_5$.  For each $a \in \mathbb{Z}_5$, state
  whether $x^2-a$ is irreducible and if not factor $x^2-a$ into
  irreducible factors.
\end{problem}

\begin{itemize}
\item $a=0$: $x^2 - 0 = (x)(x)$.

\item $a=1$: $x^2 - 1 = (x+1)(x+4) = (x+1)(x-1)$.

\item $a=2$: $x^2-2$ is irreducible.

\item $a=3$: $x^2 -3$ is irreducible.

\item $a=4$: $x^2-4 = (x-2)(x-3) = (x+3)(x+2)$.
\end{itemize}

\begin{problem} (10 points)
 Consider the Galois field $F = GF(11,x-1)$.

\begin{itemize}
\item[a.] How many elements does $F$ have?

\item[b.] Find {\em one} primitive element of $F$.

\item[c.] In terms of powers of the element you found in part (b), find
{\em all } primitive elements of $F$.
\end{itemize}
\end{problem}

a. 11 

b. 2 is primitive.

c. We know that $o(2) = 10 = |F \backslash \{0\}|$.  Then $o(2^k) =
\frac{10}{(k,10)}$.  Hence the other primitive elements are $2^3, 2^7$
and $2^9$.

\begin{problem}
(10 points)
Consider the permutation
\[
\sigma =
\left (
\begin{array}{cccccccc}
1 & 2 & 3 & 4 & 5 \\
3 & 5 & 1 & 4 & 2 
\end{array}
\right )
\]
\begin{itemize}
\item[a.] Write $\sigma$ as a product of disjoint cycles.
\item[b.] Write $\sigma$ as a product of 3-cycles.
\item[c.] What is $\sigma^{1234567}$?

\end{itemize}
\end{problem}

a. $\sigma = (1 \ 3) (2 \ 5)$

b. $\sigma = (1 \ 3 \ 5)(3 \ 5 \ 2)$

c. The order of $\sigma$ is 2 and 1,234,567 is odd, so $\sigma^{1234567}
= \sigma$.

\begin{problem}
(15 points)
Consider the dihedral group $D_3$ (the symmetries of an equilateral
triangle).  

\begin{itemize}
\item[a.] What is the order of $D_3$?

\item[b.] Let $R$ denote clockwise rotation of 120 degrees and let $F$
denote a flip about the vertical axis.  Describe all the elements of
$D_3$ and write them down in terms of $R$, $F$ and the identity.

\item[c.] For each integer $k$ between 1 and 6 (inclusive) list the
elements of $D_3$ of order $k$, or else state that no such element
exists.
\[
\begin{array}{l}
\hbox{order 1:} \\
\hbox{order 2:} \\
\hbox{order 3:} \\
\hbox{order 4:} \\
\hbox{order 5:} \\
\hbox{order 6:}
\end{array}
\]
\end{itemize}
\end{problem}

a. 6

b. identity, $R, R^2, F, RF, R^2F$.

c. Order 1: identity.  Order 2: $F$, $RF$, $R^2F$.  Order 3: $R$,
$R^2$.    No others exist.

\begin{problem} (10 points)
  Let $a$ and $b$ be two elements of a field.  Prove that $a\cdot b =
  0$ in $F$ if and only if $a$ or $b$ is zero.
\end{problem}

%\newpage
{\it Proof.} ($\Leftarrow$) is clear.

($\Rightarrow$) If $a=0$ then we are done. If $a\ne 0$ then $b=1\cdot
b=(a^{-1}a)b=a^{-1}(ab)=a^{-1}0=0$.


\begin{problem} (10 points)
  Let $p$ be a prime number and suppose that $\mathbb{Z}_p$ contains
  an element $c$ which is not a cube (in $\mathbb{Z}_p$).  Show that
  there exists a field with exactly $p^3$ elements.
\end{problem}

{\it Proof.} Let $P(x)=x^3-c$. Since $c$ is not a cube in $\Bbb{Z}_p$,
the cubic polynomial $P(x)$ is irreducible over $\Bbb{Z}_p$. The
Galois field GF$(p,P(x))$ contains exactly $p^3$ elements.



\end{document}



