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\begin{document}
\begin{center}
     {\large\bf    Review for Exam 2}
\end{center}
\begin{problems}
\item
{\bf Prove:}  If  $a,\,b\in\Z$ are relatively prime, then for every
$c\in\Z$ there exist
integers $s$ and $t$ so that $c=sa+tb$.

\item
Let $p$ be a prime number. Show that if $p$ divides the product $ab$, then
$p$ divides $a$ or
$p$ divides $b$.

\item
Find all associates of 6 in $\Z_{10}$.

\item
Find all greatest common divisors of $11+2i$ and $-7+i$  in $\Z[i]$ and give
a Bezout-equation for one of them.

\item
{\bf Prove:} If $a\not=0$ is {\em not \/}a zero-divisor in a commutative
ring $R$ and if there are $x$ and
$y$ in $R$ so that $ax=ay$, then it is $x=y$.

\item
Let $a,\,b$ be integers and $p$ be a prime number.
\\
{\bf Prove:} If $a^2\equiv b^2$ (mod $p$), then $p$ divides $a-b$ or $p$
divides $a+b$.

\item
Let $R$ be a commutative ring. Assume that for two elements $a,\,b\in R$ we
know that
$a,\,b$, and $a+b$ are all units in $R$. Show that then also
$a^{-1}+b^{-1}$ is a unit in $R$.
\\ \relax
[Hint: Calculate the fraction $\frac{1}{a}+\frac{1}{b}$.]

\item
Consider the ring $\Z_{27}$.
\begin{subproblem}
    \item  List all units.
    \item  Find the inverse of the element 16 in $\Z_{27}$ and give the
result as a number
               between 1 and 26.
    \item Calculate $\frac{13}{16}$.
    \item List the powers of 2 and find the order of the element  2.
              [Part (a) gives you a hint  for what the order of 2 could be.]
    \item What is the order of 25?
 \end{subproblem}

\item
\begin{subproblem}
\item Define carefully the notion of a zero-divisor.
\item Show that if the integers $a$ and $n$ are {\sl not\/} relatively
prime, then $a$ is a zero-divisor
          in $\Z_n$.
\item What do you know about $a$  as an element of $\Z_n$, if $a$ and $n$
are relatively prime?
\item Conclude from (a) and (b) that $a$ is a zero-divisor in $\Z_n$
exactly when $a$ and $n$ are not
         relatively prime.
\end{subproblem}


\item
Consider the ring $\Z_{37}$. The order of 2 is 36 in this ring (You don't
have to check this!).
\begin{subproblem}
\item Find an element $a$ of order $4$.
\item Give both square roots of $-1$.
\end{subproblem}


\item
\begin{subproblem}
\item  Show that the polynomial  $p(X)=3X^3+2X^2-4X+2$ has no roots in $\Z$.
\item  Show that $\frac{3}{5}$ is not a root of a monic polynomial with
integer coefficients.
\end{subproblem}

\item
Show that there exists no element $a+b\sqrt{2}\in\Z[\sqrt{2}]$ with the
property
$(a+b\sqrt{2})^k=1+\sqrt{2}$ for some $k>1$.

\item
Suppose $\Z_p$ contains a unit of order 5.
Show that $p=1+10k$ for some $k\in\Z$.
\\
The first  such prime is $p=11$.
What is the next such prime?
Can you find a unit of order 5 in the ring $\Z_p$ for your $p$?

\item
Let $a=21$ and $b=40$.
\begin{subproblem}
\item  Find $x,\,y\in\Z$ so that
\[
      x\equiv 1\ (\mbox{\rm mod } 21), \qquad x\equiv 0\ (\mbox{\rm mod }
40),\qquad
      y\equiv 0\ (\mbox{\rm mod } 21), \qquad y\equiv 1\ (\mbox{\rm mod } 40)
\]
\item Find $z\in\Z$ so that simultaneously
\[
    z=3+21k \ \;\mbox{\rm and }\  z=2+40 l
\]
for some choice of $k,\,l\in\Z$.
\end{subproblem}

\end{problems}
\end{document}
