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\noindent{\Large{\textsc{Number Theory \hfill Primitive Roots Practice Exercises\\ Fall 2023 \hfill }}}

\large

\vspace{1cm} 

\begin{exercise}
If $f(X), g(X) \in \Z[X]$ are nonzero polynomials satisfying
\[
f(X) = (X-a)g(X) + b
\]
for some $a, b \in \Z$, show that $f(X)$ and $g(X)$ have the same leading coeficient.
\end{exercise}

\begin{exercise}
If $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ are integers satisfying $0 \le a_k \le b_k$ for all $k$, show that
\[
\sum_{k=1}^n a_k = \sum_{k=1}^n b_k  \,\,\, \Rightarrow \,\,\, a_k = b_k \text{ for all } k.
\]
\end{exercise}

\begin{exercise}
Let $p$ be an odd prime.  Use the Binomial Theorem to verify the following assertions made during Monday's lecture.
\begin{itemize}

\item[\bf a.]  For any $r \in \Z$ one has $(r+p)^{p-1} \equiv r^{p-1} + (p-1)pr^{p-2} \pmod{p^2}$.

\item[\bf b.]  For any $k \ge 2$ and $a \in \Z$ one has $(1+ap^{k-1})^p \equiv 1 + a p^k \pmod{p^{k+1}}$.

\end{itemize}
\end{exercise}

\begin{exercise}
Textbook exercise 8.3.1
\end{exercise}

\begin{exercise}
Textbook exercise 8.3.3
\end{exercise}

\begin{exercise}
Textbook exercise 8.3.4
\end{exercise}

\begin{exercise}
Textbook exercise 8.3.6a
\end{exercise}

\begin{exercise}
Textbook exercise 8.3.8
\end{exercise}

\begin{exercise}
Textbook exercise 8.3.11
\end{exercise}

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