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\prof{Peter Cholak and Juan Migliore} \class{Math 222 - Sample Exam 2}
\quizdate{Friday, April 20} \pagestyle{empty}

\begin{document}


\heading

The exam will \emph{focus} on sections 6.1-6.3, 7.1-7.3, 8.2, 8.4, and
9.1-9.2 (the material we covered since the first exam)  and will have
format as outlined below.  In addition to reviewing this sample exam, it
is suggested that you review the past quizzes, the sections in the
book, your class notes and the homework.
\\

\begin{problem}(4 points each -- 5 parts) {\bf Define} the following terms:\\
  a) {\em relatively prime} polynomials\\
  b) {\em Galois imaginary}\\
  c) {\em transposition} \\
  d) {\em group}\\
  e) {\em The dihedral group $D_4$}\\
  f) {\em A ring}
\end{problem}

\begin{problem}(2 points each -- 5 parts) 
  Answer {\bf True} or {\bf False} - no work
  required:\\
  a) $x+1$ and $x-1$ are relative prime over all fields.\\
  b) $GF(3,x^5 + 2x + 1)$ contains 242 elements.\\
  c) The permutation $\begin{pmatrix}
    1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
    3 & 6 & 7 & 1 & 8 & 2 & 9 & 5 & 4
\end{pmatrix}$ is even.\\
d) The negative real numbers under multiplication is a group.\\
e) The order of the number 2 in the group $(\mathbb{Z}_3,+)$ is
2.\\
f) The set of 10th roots of unity in $\mathbb{C}$ is a group (under
multiplication).\\
g) Each element $\zeta \neq 0, 1$ in $GF(2,x^5 + x^2 + 1)$ is primitive.\\

\end{problem}

{\bf Part III:} Computational Problems - there will be a few problems
something {\em like} the following for a total of 40 points. \\

\begin{problem} 
  Completely factor $P(x) = x^3 + 2x + 3$ over $\mathbb{Z}_5$.
\end{problem}

\begin{problem} Find a greatest common divisor of $P(x) = x^5 + 4x^3 + 3x^2
  + 4x + 1$ and $Q(x) = x^4 + x^3 + 2x + 1$ over $\mathbb{Z}_5$.
\end{problem}

\begin{problem} Find the remainder when dividing $x^{51}$ by $x + 4$ over
  $\mathbb{Z}_7$.
\end{problem}

\begin{problem}
  For which $a \in \mathbb{Z}_5$ is $P(x) = x^3 + x + a$ irreducible
  over $\mathbb{Z}_5$?
\end{problem}

\begin{problem} Find a greatest common divisor of $P(x) = x^5 + 4x^3 + 3x^2
  + 4x + 1$ and $Q(x) = x^4 + x^3 + 2x + 1$ over $\mathbb{Z}_5$.
\end{problem}

\begin{problem} Find the inverse of the element $1 + 2\beta$ in $GF(3,x^2 +
  x + 2)$, where $\beta$ is the associated Galois imaginary.
\end{problem}

%\begin{problem} Let $f(x) = x^3 + x + 1$ and $\alpha$ be the associated
%  Galois imaginary in $GF(2,f(x))$.  Find all integers $i$ such that
%  $f(\alpha^i) = 0$.
%\end{problem}

\begin{problem}
  Find all the primitive elements in $\mathbb{Z}_{19}$
\end{problem}

\begin{problem} Find all primitive elements in $GF(2,x^4 + x + 1)$.
\end{problem}

\begin{problem} Let $\sigma =
\begin{pmatrix}
  1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
  3 & 6 & 7 & 1 & 8 & 2 & 9 & 5 & 4
\end{pmatrix}$
and $\rho =
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
2 & 3 & 1 & 9 & 8 & 7 & 6 & 5 & 4
\end{pmatrix}$.

a) Express both $\sigma$ and $\tau$ in disjoint cycle notation and as
a product of transpositions.

b) Express $\sigma^{-1}$ and $\sigma\tau$ in disjoint cycle notation.

c) Find the orders and parity of $\sigma$, $\tau$, and $\sigma\tau$.

d) Compute $\sigma^{34}$.
\end{problem}

%\begin{problem} Let $G = \mathbb{Q}^*$ be the nonzero rational numbers and
%  consider the following multiplication on $G$: $x*y \equiv 2xy$ where
%  multiplication is as usual on the
%  right hand side.  This forms a group.\\
  
%  a) Find the identity element, $1_G$, of $G$.
  
%  b) Find the inverse of 3 in $G$.
%\end{problem}

%\begin{problem} Consider the following hypothetical multiplication table
%  for a group $G$:\\
%\begin{tabular}{c|c c c}
%G & e & a & b\\
%\hline
%e & e & a & b\\
%a & a & e & a\\
%b & b & a & e\\
%\end{tabular}\\

%The table defines a multiplication, but the nature of the table should
%suggest that $G$ is in fact {\em not} a group.  Which of the
%properties of a group fail?  Which hold.  Explain.
%\end{problem}

\begin{problem} Consider the following set of matrices:
  $$
  G \equiv GL_2(\mathbb{R}) = \{ A =
\begin{pmatrix}
  a & b\\
  c & d\\
\end{pmatrix}: a, b, c, d \in \mathbb{R} \text{ and } \operatorname{det}(A)
\neq 0\}
$$
$G$ is a group under matrix multiplication - called the general
linear group of 2 by 2 invertible matrices over the real numbers.  Why
is $G$ closed under multiplication? What is the identity element? Why?
Why does each matrix have an inverse?  What is the inverse of the
element $\begin{pmatrix}
  2 & 4\\
  1 & 3\\
\end{pmatrix}$?
\end{problem}

\begin{problem} 
  How many elements of order 2 are there in $D_{125}$? Explain.
\end{problem}

\begin{problem}
  What does $R_3FR_2$ equal in $D_4$?  What is the inverse of $R_2F$
  in $D_4$?
\end{problem}

%\begin{problem}
%  Identify all the subgroups of $S_3$.
%\end{problem}


%\begin{problem} Consider the map $f: (\mathbb{Z}_8,+) \to 
%  (\mathbb{Z}_8,+)$ by $f(x) = 4x$.
%  Is this an isomorphism of groups? Why or why not? % Consider the set
%  %$S = \{x \in \zz_8: f(x) = 0\}$.  How many elements are in $S$?  Is
%  %it a subgroup of $(\zz_8,+)$.  Why or why not?  If so, is it normal?
%\end{problem}


%\begin{problem} Which of these maps of groups are isomorphisms? Why?

%a) $f: (\mathbf{C},+) \to (\mathbf{C},+)$ by $f(a + bi) = a - bi$.

%b) $f: (\mathbf{C}^*,\cdot) \to (\mathbf{C}^*,\cdot)$ by $f(a + bi) = a - bi$.

%c) $f: (\mathbb{Q}^*,\cdot) \to (\mathbb{Q}^*,\cdot)$ by $f(x) = 1/x$.

%d) $f: (\mathbb{R},+) \to (\mathbb{R},+)$ by $f(x) = |x|$.
%\end{problem}

{\bf Part IV:} Proofs - {\em exactly} two of the following problems
will appear.
They'll be worth 15 points each.\\

\begin{problem} 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}

\begin{problem} 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}

\begin{problem} Let $G$ be a group.  Prove that if $x^2 = 1$ for each $x
  \in G$, then $G$ is abelian.
\end{problem}

%\begin{problem}  
%  Let $h$ be a function from a group $(G,\cdot)$ to a group
%  $(H,\cdot)$ such that for any $a,b \in G$, $h(ab)=h(a)h(b)$.  The
%  kernel of $h$ is defined to be $\{g \in G : h(g) = 1_H\}$.  Show
%  that the kernel of $h$ is a subgroup of $G$.
%\end{problem}

\begin{problem}
  Let $f: X \to Y$ be a function, where $X$ and $Y$ are finite sets of
  the same size.  Show that $f$ is onto iff $f$ is 1-1.
\end{problem}

\begin{problem}
  Let $F = GF(p,P(x))$ where $P(x)$ is irreducible.  Show that for any
  non-zero $a$ in $F$ there at most one $p$th root of $a$.
\end{problem}

\begin{problem}
Prove that if $n \geq 2$ then every permutation of $\{ 1,2,\dots,n\}$
can be expressed as the composition of transpositions of the form $(1 \
a)$, $a = 2,\dots,n$.
\end{problem}

\end{document}


