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\begin{document}
\junk{
{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Heide Gluesing-Luerssen

\begin{center}
{\bf Mathematics 222.02: Algebraic Structures }
\\
{\bf Spring Semester 1998}
\\
{\bf Exam 2}
\\
{\bfMarch 6, 1998}
\end{center}
%\vspace*{.4cm}
\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm}
Examination contains 7 problems on 6 sheets of paper
including the front cover.
Do all your work in this booklet and show your computations.
Calculators, books, and notes are not allowed.
\\[1.7ex]
\hline
\end{tabular}
\begin{center} {\bf Scores}
\\[1.5ex]
\begin{tabular}[t]{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
  1      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  2      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  3      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  4      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  5      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  6      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  7      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
Total    & 100\mbox{\  \ } & \rule[-.5cm]{0cm}{1.2cm} \\ \hline
\end{tabular}
\end{center}
\vspace*{1.4cm}
{\bf Sign the pledge:}
\\[1ex]
``On my honor, I have neither given nor received unauthorized aid on this
Exam.''

{\bf Signature:} \underline{\mbox{\hspace*{7cm}}}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK}\end{center}
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\newpage
}
\begin{center}
     {\large\bf    Some Problems for Exam 2 and  Review 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
{\em One for practice, one for the exam.}
\begin{subproblem}
\item  Find all greatest common divisors of $11-10i$ and $-13i$ in $\Z[i]$
and give
          a Bezout-equation for one of them.
\item The same for $11+2i$ and $-7+i$.
\end{subproblem}

\item
{\bf Prove:} if $a$ 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.
    \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 $\Z_{41}$.
\begin{subproblem}
\item Assume that $a\not=1$ is an element of $\Z_{41}$  with $a^{15}=1$.
          Show that the order of $a$ is 5.
\item Find the order of 2. [Make sure that you get a divisor of 40.]
\item Find four elements of order 5 in $\Z_{41}$.
\item Use the list you produced in (b) to find a square root of $-1$ in
$\Z_{41}$.
\item How many square roots of 2 does $\Z_{41}$ have?  (You don't have to
find any!)
\item Suppose $a\in\Z_{41}$ is a square root of 2, i.~e. $a^2=2$.
          Show that 20 divides  the order of $a$.
\item Show that $a^{20}\not=1$ and use this to conclude that the order of
$a$ has to be 40.
\item Check that $17^2=2$.
\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
Let $a$ and $b$ be relatively prime numbers.
Assume that there are numbers $x$ and $y$ so that
\[
        ax\equiv ay \text{ (mod $b$).}
\]
Show that it follows $x\equiv y$ (mod $b$).

\item
\begin{subproblem}
\item Find a Bezout-equation for $a=8$ and $b=35$.
\item Use (a) to find a solution to the equation $8X\equiv 7$ (mod 35).
\end{subproblem}

\item
\begin{subproblem}
\item Show that the polynomial $p(X)=2X^2+3X-7$ has no roots in $\Z$.
\item the same for $p(X)=3X^3+2X^2-4X+2$.
\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 $k>1$.



\end{problems}
\end{document}
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 ------------------------------------------------------------
 | Heide Gluesing-Luerssen   |Phone (office):  219-631-8698 |
 | Department of Mathematics |Phone (home):    219-243-2609 |
 | University of Notre Dame  |Fax:             219-631-6579 |
 | Notre Dame,  IN 46556     |http://www.nd.edu/~hgluesin/  |
 | USA                       |                              |
 ------------------------------------------------------------

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