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\textbf{18.782 Introduction to Arithmetic Geometry\hspace{118pt}Fall~2020}\\\vspace{4pt}
\textbf{Problem Set \#4\hfill Due: 10/02/2020}
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These problems are related to the material covered in Lectures 8-9.
I have made every effort to proof-read them, but there are may be errors that I have missed.
The first person to spot each error will receive 1-5 points of extra credit.
\medskip

The problem set is due by 5pm on 10/02/2020 and should be submitted electronically as a pdf-file e-mailed to  \url{zzyzhang@mit.edu} and \url{roed@mit.edu} (please include ``18.782'' in the subject of the email).
You can use the latex source for this problem set as a template for writing up your solutions;
be sure to include your name in your solutions and remember to identify all collaborators and any sources that you consulted that are not listed in the syllabus.

\subsection*{Problem 1.  A stronger form of Hensel's lemma. (30 points)}

\begin{enumerate}
\item[\textbf{(a)}] Let $f\in\Zp[x]$ and suppose $\lvert f(a) \rvert_p < \lvert f'(a) \rvert_p^2$ for some $a\in\Zp$.
Let $a_1=a$, and for $n\ge 1$ let
\[
a_{n+1}=a_n-f(a_n)/f'(a_n).
\]
Prove that this defines a Cauchy sequence $(a_n)$ in $\Zp$ whose limit $b$ uniquely satisfies $f(b)=0$ and $|a-b|_p < |f'(a)|_p$, and moreover, $|f'(a)|_p=|f'(b)|_p$.
 (you may find it helpful to reword this in terms of $v_p$ and work with congruences modulo powers of $p$).

\item[\textbf{(b)}] Prove that the hypothesis in (a) is necessary in the following sense.
Suppose that $b$ is a simple root of a polynomial $f\in\Zp[x]$.  Prove that for any $a\in\Zp$, if $|a-b|_p < |f'(b)|_p$ then $|f(a)|_p<|f'(a)|_p^2$.  Conclude that if no $a\in\Zp$ satisfies the hypothesis of (a), then $f(x)$ does not have a simple root in $\Zp$.


\item[\textbf{(c)}] Use (a) to compute a square root of 57 in $\Z_2$ to 16 digits of 2-adic precision using $a=1$.
How many $a_n$ do you need to compute to achieve this precision?
\end{enumerate}

\subsection*{Problem 2. A faster form of Hensel's lemma. (20 points)}

\begin{enumerate}
\item[\textbf{(a)}] Let $R$ be a commutative ring, let $f\in R[x]$, and let $m\in R$.
Suppose that $x_0,z_0\in R$ satisfy $f(x_0)\equiv 0\bmod m$ and $f'(x_0)z_0\equiv 1\bmod m$ (note that $a\equiv b\bmod m$ simply means that $a-b$ is an element of the $R$-ideal $(m)$).
Let
\begin{align*}
x_1&=x_0-f(x_0)z_0,\\
z_1&=2z_0-f'(x_1)z_0^2.
\end{align*}
Prove that
\begin{enumerate}[(i)]
\item $x_1\equiv x_0\bmod m$,
\item $f(x_1)\equiv 0\bmod m^2$,
\item $f'(x_1)z_1\equiv 1\bmod m^2$,
\end{enumerate}
and that (i) and (ii) uniquely characterize $x_1$ modulo $m^2$.
\item[\textbf{(b)}] Use part (a) to compute a cube-root of 9 in the ring $\Z_{10}$ to 64 digits of $10$-adic precison by working modulo $10, 10^2, 10^4, 10^8, 10^{16}, 10^{32}, 10^{64}$.
\item[\textbf{(c)}] Prove that Fermat's last theorem is false in $\Z_{10}$.
\end{enumerate}

\subsection*{Problem 3. Applications of Hensel's lemma (50 points)}
Recall that every element of $\Qp^\times$ can be uniquely written as $p^ru$ with $r\in\Z$ and $u\in\Zp^\times$.
Let $\Qp^{\times n}=\{x^n:x\in\Qp\}$ denote the set of $n$th powers in $\Qp^\times$.

\begin{enumerate}
\item[\textbf{(a)}] For all odd primes $p$, prove that $p^ru$ is a square in $\Qp^\times$ if and only if $r$ is even and $u$ is a square modulo $p$.  Conclude that $\Qp^\times/\Qp^{\times 2} \simeq (\Z/2\Z)^2$ (as finite abelian groups).\footnote{Anytime $\Z/p\Z$ (or any ring for that matter) appears in a context where a group is required, you can assume it is the additive group that is being referred to (one uses $(\Z/p\Z)^\times$ for the multiplicative group).}

\item[\textbf{(b)}] Using the strong form of Hensel's lemma, prove that $2^ru$ is a square in $\Q_2^\times$ if and only if $r$ is even and $u\equiv 1\bmod 8$.
Conclude that $\Q_2^\times/\Q_2^{\times 2}\simeq (\Z/2\Z)^3$.
\item[\textbf{(c)}] Determine the structure of $\Qp^\times/\Qp^{\times n}$ for all primes $p$ and odd primes $n$.
\end{enumerate}
Let $\mu_{n,p}=\{x\in\Q_p:x^n=1\}$ denote the set of $n$th roots of unity in $\Q_p$.
\begin{enumerate}
\item[\textbf{(d)}] Prove that $\mu_{n,p}$ is a subgroup of $\Z_p^\times$.
\item[\textbf{(e)}] Use Hensel's lemma to prove that for $p\nmid n$ the group $\mu_{n,p}$ is cyclic of order $\gcd(n,p-1)$.
\item[\textbf{(f)}] Let $p$ be odd.  Use the strong form of Hensel's lemma to prove that $\mu_{p,p}$ is trivial.  Conclude that there are exactly $p-1$ roots of unity in $\Qp$ (be sure to address $\mu_{p^r,p}$).
\item[\textbf{(g)}] Prove that $\mu_{4,2}=\mu_{2,2}=\{\pm 1\}$.  Conclude that $\pm 1$ are the only roots of unity in $\Q_2$.
\end{enumerate}


\subsection*{Problem 4. Survey}
Complete the following survey by rating each problem on a scale of 1 to~10 according to how interesting you found the problem (1 = ``mind-numbing," 10 = ``mind-blowing"), and how difficult you found the problem (1 = ``trivial," 10 = ``brutal").  Also estimate the amount of time you spent on each problem.

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\begin{tabular}{l|r|r|r|}
& Interest & Difficulty & Time Spent\\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
Problem 3 & & & \\\hline
\end{tabular}
\end{center}

\noindent
Feel free to record any additional comments you have on the problem sets or lectures; in particular, how you think they might be improved.

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