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\begin{document}
\begin{center}
     {\large\bf    Review for Exam 3}
\end{center}
\begin{problems}
\item
Let $p$ and $q$ be positive prime numbers.
List all positive divisors of $pq^3$.

\item Write $-525$ as a product of prime numbers.

\item
Given $k$ different prime numbers $p_1,\ldots,\,p_k$.
Construct a prime number which is not on this list.

\item
Write both $3-4i$ and $6+9i$ as products of Gaussian primes.
Show also that the two given Gaussian integers are relatively prime.

\item
Give all associates of 3 in
\begin{subproblem}
\item in $\Z$,
\item in $\Z[i]$,
\item in $\Z[\sqrt{2}]$.
\end{subproblem}

\item
Factorize  61 into primes in the ring
\begin{subproblem}
\item $\Z$,
\item $\Z[i]$,
\item $\Z[\sqrt{2}]$.
\end{subproblem}

\item
Factorize $28+16i\in\Z[i]$ into Gaussian primes.

\item
Let $p\in\Z$ be a prime number and $\alpha,\,\beta\in\Z[i]$ be Gaussian
integers with $\beta$
not being a unit.
{\bf Prove:}  If $\alpha\beta=p^2$,  then $N(\alpha)=p^k$ for some integer
$k\leq 3$.

\item
For all of the following $p\in\Z$ state whether $p$ is a Gaussian prime or
not.
If not, give a factorization of $p$ into two non-units.

{\bf (a)} $p=31$\hspace*{1cm}
{\bf (b)} $p=33$ \hspace*{1cm}
{\bf (c)} $p=2$\hspace*{1cm}
{\bf (d)} $p=101$ \hspace*{1cm}
{\bf (e)} $p=103$.


\item
Consider the symmetric group $S_7$.
\begin{subproblem}
\item Give all elements of $S_7$, which are in $D_7$, and represent them as
product
          of disjoint cycles. Give also the sign ${\rm sgn}\;\sigma$ of all
these elements.
\item Can you give a way to write $S_3$ as a subgroup of $S_7$?
\item Give an element of order $7$ in $S_7$.
\item Give a subgroup of order 4.
\end{subproblem}

\item
A $k$-cycle is a cycle of length $k$, i.~e. a permutation of the form
$(a_1\;a_2\;\ldots a_k)$ with
different $a_i$.
{\bf Prove:}
A $k$-cycle is its own inverse exactly when $k=2$.


\item
Let $\sigma=(2\,5\,3)$ and $\tau=(1\,2\,4)$ in $S_5$.  Calculate
\begin{subproblem}
\item $\tau\sigma$,
\item $\sigma\tau\sigma^{-1}$,
\item  $\sigma^{17}$,
\item find $\alpha\in S_5$ so that $\alpha\sigma(1\,4)=(1\,2\,3)$.
\end{subproblem}

\item
{\bf Prove:} If $G\subseteq S_n$ is a subgroup containing all permutations
then $G= S_n$.
%{\bf Prove:}  If $G\subseteq S_5$ is a subgroup of $S_5$ containing $(1\,
2)$ and $(1\,2\,3\,4\,5)$,
%then $G=S_5$.

\item
In $\Z[\sqrt{2}]$ the following equation is true
\[
     (5+2\sqrt{2})(-3+5\sqrt{2})=5+19\sqrt{2}=(7-4\sqrt{2})(11+9\sqrt{2}).
\]
\begin{subproblem}
\item Calculate the norms of all factors given in the products on the left
and on the right and show that these
         factors are prime elements in $\Z[\sqrt{2}]$.
\item How does the result of (a) relate to the fact that $\Z[\sqrt{2}]$ is
a unique factorization domain?
          Give the exact relationship between the two factorizations.
\end{subproblem}

\item
Consider the ring $\Z[2i]=\{a+2bi\,|\,a,\,b\in\Z\}$, which can be regarded
as a subset of $\Z[i]$.
\begin{subproblem}
\item  Determine the units of $\Z[2i]$,
\item Show that $\Z[2i]$ is not a unique factorization domain by giving two
appropriate factorizations of
          $4\in\Z[2i]$.
\end{subproblem}

\item
{\bf Prove by induction:}
$5$ divides $3^{4n}-1$ for all $n\geq1$.


\end{problems}
\end{document}
