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\def\Z{{\mathbb Z}}
\def\Q{{\mathbb Q}}
\def\R{{\mathbb R}}
\def\lra{\longrightarrow}

\centerline{\bf Mathematics 361:  Number Theory}
\centerline{\bf Assignment D}

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{\bf Reading:  }Ireland and Rosen, Chapter 4 (including the exercises)
and into Chapter~5

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For this assignment it will be very helpful to bear in mind the
following result:
\begin{quote}
Given $n$ and~$e$, let $\tilde e=\gcd(e,n)$.
The multiplication-by-$e$ map $\Z/n\Z\lra\Z/n\Z$ given by $x\mapsto
ex$ has image $\langle\tilde e+n\Z\rangle$ of order $n/\tilde e$,
and it has kernel $\langle n/\tilde e+n\Z\rangle$ of order~$\tilde e$,
so it is $\tilde e$-to-$1$.  Especially the map is an isomorphism if
$\gcd(e,n)=1$.
\end{quote}
This result transfers to the multiplicative group $G=(\Z/p\Z)^\times$
where $p$ is prime, a cyclic group having a generator~$g$, through the
isomorphism $(\Z/(p-1)\Z,+)\lra(G,\cdot)$ given by $a\mapsto g^a$.  The
self-map of~$G$ corresponding to $x\mapsto ex$ on $(\Z/(p-1)\Z,+)$
is $y\mapsto y^e$.
Thus the $e$th power map on~$(\Z/p\Z)^\times$ is is $\tilde e$-to-$1$
and it takes the $(p-1)/\tilde e$ values
$\{1,h,h^2,\dotsc,h^{(p-1)/\tilde e-1}\}$ where $h=g^{\tilde e}$.

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{\bf Problems:}

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Ireland and Rosen, Exercises 4.8, 4.13 as it {\em should\/} be phrased
(do these first and then use them freely in working the rest of the
problems); 4.1, 4.17, 4.18; 4.2 (for $p=7,11,13$), 4.19; 4.10
(let $f(d)=\sum_{u:\text{ order }d}u$ and let $g(d)=\sum_{u:u^d=1}u$,
which can be evaluated as a geometric sum; $g$ has an expression in
terms of~$f$ and then M\"obius inversion gives $f$ in terms of~$g$;
the exercise is requesting $f(p-1)$); 4.20 excluding the $p=19$ part.
(The more you use algebra, the less tedious these will be.)

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