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\prof{Chris Bendel and Peter Cholak} \class{Math 222 - Sample Exam 2}
\quizdate{Wednesday, April 14} \pagestyle{empty}

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\heading

The exam will cover sections 6.1-6.3, 7.1-7.3, 8.2, 8.4, 9.1, 9.2 and will
have format as
outlined below.\\

\begin{problem}(4 points each) {\bf Define} the following terms:\\
a) {\em relatively prime} polynomials\\
b) {\em field}\\
c) {\em Gauss imaginary}\\
d) {\em transposition}
\end{problem}

\begin{problem}(2 points each) Answer {\bf True} or {\bf False} - no work
required:\\
a) The polynomial $P(x) = 2x^3 + x^2 + 3$ is irreducible over $\mathbb{Z}_5$.\\
b) $GF(3,x^5 + 2x + 1)$ contains 242 elements.\\
c) The negative real numbers under multiplication is a group.\\
d) 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.
\end{problem}
\\
{\bf Part III:} Computational Problems - there will be a few problems
something {\em like} the
following for a total of 45 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 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 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 $\sigma\tau$.
\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.\\

\newpage
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}

{\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
$p^3$-elements.
\end{problem}

\begin{problem} Let $p$ be a prime number and $P(x)$ be an irreducible
polynomial of degree
$\nu$ over $\mathbb{Z}_p$.  Suppose that $n$ is a positive integer
relatively prime to
$p^{\nu} - 1$.  Prove that there is exactly one $n$th root of unity in
$GF(p,P(x))$.
\end{problem}

\begin{problem} Let $\alpha \in GF(p,P(x))$ be primitive.  Show that
$\alpha^2$ is primitive
if and only if $p = 2$.
\end{problem}

\begin{problem} Let $a, b, c$ be elements of a group $G$.  Prove that if
$a\cdot b = a\cdot
c$, then $b = c$.  Can we make the same conclusion if $b\cdot a = a\cdot
c$?  Why or why not?
\end{problem}

\begin{problem} Let $a$ and $b$ be elements of a group $G$.  Prove that if
$(a\cdot b)^2 =
a^2\cdot b^2$, then $a\cdot b = b\cdot a$.
\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 $G$ be a group.  Prove that if $a\cdot b = 1_G$ for
some $a, b \in G$,
then $b = a^{-1}$.
\end{problem}


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