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\begin{document}
\begin{center}
     {\large\bf    Review for the Final Exam in Math 222.02}
\end{center}
\begin{problems}
\item
{\bf Prove} that $\sqrt{7}$ is irrational.

\item
Make sure that you know how to calculate in $\Z,\,\Z[i],\,\Z[\sqrt{2}]$.
\\[1ex]
For example: Given $-1+5\sqrt{2},\,16+11\sqrt{2}\in\Z[\sqrt{2}]$.
\begin{subproblem}
\item Calculate a greatest common divisor.
\item List all greatest common divisors.
\item Find a B\'ezout-equation.
\end{subproblem}

\item
Show all associates of $2+3i\in\Z[i]$ in the Gaussian plane.

\item
Find all roots of the polynomial $f=X^2+X$ in $\Z_{10}$.

\item
Let $\sigma,\,\tau\in S_n$ for some $n\geq2$.
What is the inverse of the element $\sigma^2\tau^3\sigma^{-1}$?

\item
{\bf Prove:}  If $b$ is a unit of a ring $R$, then $b$ is not a zero divisor.

\item
Given the polynomials $f=X^4+5X^2+3X+1$ and $g=X^2+2X+3$ with coefficients
in $\Z_9$.
Find polynomials $a$ and $r$ so that $f=ag+r$ and $\deg r<\deg g$.

\item
Can you find a number $a+b\sqrt{2}\in\Z[\sqrt{2}]$ so that
$(a+b\sqrt{2})(2-\sqrt{2})=1$?

\item
Find ${\Ds \frac{2}{5}\in\Z_{13}}$ and ${\Ds \frac{4}{9}\in\Z_{10}}$.

\item
Give an example of a ring $R$ and two units $a,\,b\in R$, so that $a+b$ is
not a unit.

\item
\begin{subproblem}
\item Determine the order of 2 in $\Z_{17}$.
\item Why does $\Z_{17}\backslash\{0\}$ together with multiplication
(modulo 17) form a
          group?
\item Give a subgroup of order 4.
\item Is 2 a square in $\Z_{17}$?
\item Is -1 a square in $\Z_{17}$?
\end{subproblem}

\newpage
\item
Factorize into Gaussian primes:
\begin{subproblem}
\item $35i$,
\item $-31$,
\item $50+75i$,
\item $3+15i$.
\end{subproblem}

\item
Consider the symmetric group $S_6$ and its subgroup $D_6$ of all symmetries
of the regular
hexagon.
\begin{subproblem}
\item Find a subgroup of order 3 of $D_6$ and describe its action on the
hexagon.
\item List all flips of $D_6$ which have positive sign.
\item Find the smallest subgroup of $D_6$ containing the rotation
$(1\,2\,3\,4\,5\,6)$ and
          $(1\,3)(4\,6)$.
\end{subproblem}

\item
Consider the group $\Z_{18}$ with the addition $+$.
\begin{subproblem}
\item Find all generators.
\item Determine the order of the element 12.
\item Give a subgroup of order 6 and determine all its cosets.
\end{subproblem}

\item
{\bf Prove}: If $G$ is a group with prime order $p=|G|$, then the only
subgroups of $G$ are
$\{e\}$ and $G$.




\end{problems}
\end{document}
