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\section*{Math 3140: Homework 2}




\begin{enumerate}
\item[A.]
\begin{enumerate}
\item[3.1]  Show that each of the following collections of numbers forms a group under addition.
\begin{enumerate}
\item[(i)] The even integers.
\item[(ii)] All real numbers of the form $a+b\sqrt{2}$, where $a,b\in \ZZ$.
\item[(iii)] All real numbers of the form $a+b\sqrt{2}$, where $a,b\in \QQ$.
\end{enumerate}
\item[3.5] Let $n$ be a positive integer.  \emph{Multiplication modulo $n$} is the function
$$\begin{array}{rccc} \cdot_n: & \ZZ\times \ZZ & \longrightarrow & \ZZ_n\\
& (a,b) & \mapsto & a\cdot_n b,
\end{array}
$$
where $ab=mn+a\cdot_n b$ for some $m\in \ZZ$.

 Prove that
$$(x\cdot_n y)\cdot_n z=x\cdot_{n}(y\cdot_n z),$$
for all $x,y,z\in \ZZ$.
\item[3.9] Let $p$ be a prime number and let $x$ be an integer which satisfies $1\leq x\leq p-1$.  Show that none of $x,2x,\ldots, (p-1)x$ is a multiple of $p$.  Deduce the existence of an integer $z$ such that $1\leq z\leq p-1$ and $x\cdot_p z=1$.
\item[3.10]  Use the results of 3.5 and 3.9 to verify that multiplication modulo $n$ makes $\{1,2,\ldots,n-1\}$ a group if $n$ is prime. 
\end{enumerate}
\item[B.]  4.1, 4.4, 4.5, 4.6, 4.8
\end{enumerate}

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