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\begin{flushright}
  Your Name Here
\end{flushright}


\begin{center}
  \textbf{Math 71}\\ Homework 1
\end{center}



\begin{itemize}
\item [\S0.1: \#5] Determine whether the following functions are
  well-defined:
  \begin{enumerate} [label=(\alph*)]
  \item $f:\Q \mapsto \Z$ defined by $f(a/b) = a$.

  \item $f:\Q \mapsto \Q$ defined by $f(a/b) = a^2/b^2$.

  \end{enumerate}

  \begin{solution}
    Here is the solution to the first problem.
  \end{solution}


\item [\S0.1: \#7] $f:A \mapsto B$ is a surjective map of sets.  Show
  $a \sim a' \Leftrightarrow f(a) = f(a')$ defines an equivalence
  relation on $A$ whose equivalence classes are the fibers of $f$.

 
\item [\S0.2: \#7] Show $\sqrt p \not\in \Q$.

 
\item [\S0.3: \#11]Show that if $\bar a, \bar b \in (\Z/n\Z)^\times$,
  then $\bar a \cdot \bar b \in (\Z/n\Z)^\times$.

  

\item [\S0.3: \#12] Suppose $n > 1$ is an integer and $a \in \Z$ with
  $1 \le a \le n$, and further that $d = (a,n) > 1$. Show there exists
  an integer $b$ with $1 \le b < n$ so that $ab \equiv 0\pmod n$and
  deduce that there cannot be an integer $c$ with
  $ac \equiv 1 \pmod n$.


\item [\S0.3: \#13]Suppose $n > 1$ is an integer and $a \in \Z$ with
  $1 \le a \le n$. If $(a,n) =1$, show that there is an integer $c$
  with $ac \equiv 1 \pmod n$.


\item [\S0.3: \#14] Show that
  $(\Z/n\Z)^\times = \{\bar a \in (\Z/n\Z) \mid (a,n) = 1 \}$.


\item [\S1.1: \#24] If $a, b$ are commuting elements of $G$, prove
  that $(ab)^n = a^nb^n$ for all $n \in \Z$.

  
\item [\S1.1: \#34] If $x$ is an element of infinite order in $G$,
  prove that the elements $x^n$, $n \in \Z$ are all distinct.



\item [\S1.1: \#35] If $x$ is an element of finite order $n$ in $G$,
  use the division algorithm to show that any integral power of $x$
  equals one of the elements in the set
  $\{1, x, x^2, \dots, x^{n-1}\}$.

  

\end{itemize}

\end{document}


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