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\textbf{18.782 Introduction to Arithmetic Geometry\hspace{118pt}Fall~2020}\\\vspace{4pt}
\textbf{Problem Set \#3\hfill Due: 9/25/2020}
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These problems are related to the material covered in Lectures 5-7.
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 9/25/2020 and should be submitted electronically as a pdf-file e-mailed to  \url{zzyzhang@mit.edu} and \url{roed@mit.edu}.
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.  Non-archimedean topology. (35 points)}
Let $k$ be a field with a non-archimedean absolute value $\Vert\ \Vert$.
The metric $d(x,y)=\Vert x - y\Vert$
makes $k$ a metric space whose topology is defined by the basis of open sets consisting of all \emph{open balls} $B(x,r)=\{y:d(x,y)<r\}$ with $x\in k$ and $r\in \R_{> 0}$.

Non-archimedean topology can be very counterintuitive, so be sure to keep in mind the following definitions, which apply to any topological space.
\begin{enumerate}
\item The collection of open sets is closed under arbitrary unions (including infinite unions) and finite intersections, and always includes the empty set and the whole space. 
\item A set is closed if and only if its complement is open.
\item The \emph{interior} of a set $S$ is the union of all the open sets $S$ contains, and the \emph{closure} of $S$ is the intersection of all the closed sets that contain $S$.
\end{enumerate}

\begin{enumerate}
\item [\textbf{(a)}] Prove that every triangle in $k$ is either equilateral, or acute isosceles. More precisely, for any $x,y,z\in k$ prove that the three distances $d(x,y), d(y,z), d(z,x)$ are either all equal, or two of them are equal and the third is smaller.
\item [\textbf{(b)}] Prove that every point in an open ball $B(x,r)$ is ``at the center" by showing that $y\in B(x,r)$ implies $B(x,r)=B(y,r)$.  Conclude that two balls are either disjoint or concentric (this means they  have a common center).
\item[\textbf{(c)}] Prove that every open ball is closed (and therefore equal to its closure).
\item[\textbf{(d)}] Consider a \emph{closed ball} $C(x,r)=\{y:d(x,y)\le r\}$.  Prove that every point in $C(x,r)$ is at the center, and that every closed ball is open (and therefore equal to its interior).
\item[\textbf{(e)}] Consider a \emph{sphere} $S(x,r)=\{y:d(x,y)=r\}$.
Prove that every sphere is both open and closed.
\item[\textbf{(f)}] Prove that $k$ is totally disconnected.  This means that for all distinct $x,y\in k$ there are disjoint open sets $X$ and $Y$ containing $x$ and $y$ respectively, such that $k=X\bigcup Y$.
\item[\textbf{(g)}] For $k=\Qp$ with the $p$-adic absolute value, prove that the closure of the open $B(x,r)$ is not necessarily the closed ball $C(x,r)$, but sometimes is (give examples of both).
Prove that the total number of open balls is countable and that every open ball has uncountably many distinct radii (that is, $B(x,r)=B(x,r')$ for uncountably many $r'$).
\end{enumerate}

\subsection*{Problem 2. $n$-adic rings (35 points)}
For any integer $n>1$ define the \emph{$n$-adic valuation} $v_n(x)$ of nonzero $x\in\Q$ to be the unique integer~$k$ for which
$x=\frac{a}{b}n^k$, with $n\nmid a$, $\gcd(a,b)=1$ and $\gcd(b,n)=1$, and let $v_n(0)=\infty$.
Now define the function $|\  |_n\colon \Q\to \R_{\ge 0}$ by
\[
| x|_n = n^{-v_n(x)},
\]
where $|0|_n=n^{-\infty}$ is understood to be $0$.
\begin{enumerate}
\item[\textbf{(a})] Prove that $|\ |_n$ is an absolute value if and only if $n$ is prime, but that $|\ |_n$ always satisfies the non-archimedean triangle inequality $| x+y|_n \le \max(| x|_n,| y|_n)$.
\end{enumerate}
Let $A_k=\Z/n^k\Z$ and consider the inverse system of rings $(A_k)$ with morphisms $A_{k+1}\to A_k$ given by reduction modulo $n^k$.
Define the \emph{ring of $n$-adic integers} as the inverse limit $\Z_n=\varprojlim A_k$ (this generalizes our definition of $\Z_p$).
\begin{enumerate}
\item[\textbf{(b)}] Prove that if $n$ is not a prime power then $\Z_n$ is not an integral domain. \end{enumerate}
In view of (b), we cannot define $\Q_n$ as the fraction field of $\Z_n$ in general.  But we can define $\Q_n$ as the completion of $\Q$ with respect to $|\ |_n$, in the same way that we constructed $\Q_p$ as the $p$-adic completion of $\Q$ (giving us an alternative definition of $\Q_p$).

\begin{enumerate}
\item[\textbf{(c)}] Formalize this definition of $\Q_n$ and show that it is a ring.
Make it clear what the elements of $\Q_n$ are, define the ring operations, and extend the definition of $|\ |_n$ to $\Q_n$.
\item[\textbf{(d)}] Which of parts $(a)$ to $(g)$ in Problem 1 still apply if we replace $k$ with $\Q_n$ and use the metric $d(x,y)=|x-y|_n$?
\item[\textbf{(e)}] Prove that for all prime powers $p^e$ we have $\Q_{p^e}=\Q_p$.

\item[\textbf{(f)}] Prove that if $p_1,\ldots p_r$ are the distinct prime divisors of $n$ then $\Q_n$ is isomorphic to the direct product of rings $\Q_{p_1}\oplus\cdots\oplus\Q_{p_r}$ and is not a field unless $r=1$.
\end{enumerate}


\subsection*{Problem 3. Quadratic extensions of $\Q_p$ (30 points)}
 Let $p$ be a prime congruent to 3 mod 4.
\begin{enumerate}
\item[\textbf{(a)}] Prove that $-1$ does not have a square-root in $\Q_p$ and that the ideal $\p=(p)$ is prime in $\Z[i]$, the ring of integers of $\Q(i)$.
\item[\textbf{(b)}] Consider the quadratic extension $\Q_p(i)=\Q_p[x]/(x^2+1)$ and
extend the $p$-adic absolute value $|\  |_p$ from $\Q_p$ to $\Q_p(i)$ by defining for each $\alpha=a+bi\in\Q_p(i)$:
\[
|\alpha|_p = |N_{\Q_p(i)/\Q_p}(\alpha)|_p^{1/[\Q_p(i):\Q_p]} = \sqrt{|a^2+b^2|_p}.
\]
Prove that $|\ |_p$ is an absolute value on $\Q_p(i)$, and that $\Q_p(i)$ is complete with respect to this absolute value.
\item[\textbf{(c)}] For $\alpha\in\Q(i)$ define
\[
|\alpha|_\p = N_{\Q(i)/\Q}(\p)^{-v_\p(\alpha)} = p^{-2v_\p(\alpha)},
\]
where $v_\p(\alpha)$ is the $\p$-adic valuation of $\alpha$.
Recall that the $\p$-adic valuation of $\beta\in\Z[i]$ is the
exponent of $\p$ in the prime factorization of the $\Z[i]$-ideal $(\beta)$,
and this extends to $\Q(i)$ via $v_\p(\beta/\gamma)=v_\p(\beta)-v_\p(\gamma)$ (note that $\Q(i)$ is the fraction field of its ring of integers $\Z[i]$).
Let $\Q(i)_\p$ denote the completion of $\Q(i)$ with respect to~$|\ |_\p$.
Prove that $\Q_p(i)$ and $\Q(i)_\p$ are isomorphic fields with equivalent (but not equal) absolute values.

\item[\textbf{(d)}] Prove that $p$ does not have a square root in $\Q_p(i)$, thus the quadratic extensions $\Q_p(\sqrt{p})$ and $\Q_p(i)$ are distinct.  Conclude that $\Q_p(\sqrt{-p})$ is also a quadratic extension of $\Q_p$, and it is distinct from both $\Q_p(i)$ and $\Q_p(\sqrt{p})$.
\end{enumerate}

We will see that up to isomorphism,  $\Q_p(i)$, $\Q_p(\sqrt{p})$ and $\Q_p(\sqrt{-p})$ are the only quadratic extensions of $\Q_p$, and that a similar statement holds for primes that are congruent to $1\bmod 4$.
This is in contrast to $\Q$, which has infinitely many non-isomorphic quadratic extensions, and to $\Q_\infty=\R$, which has only one.

\begin{enumerate}
\item[\textbf{(e)}] Recall that every field $K$ with a nonarchimedean absolute value has an associated residue field, defined by $\OK / \mK$, where
\begin{align*}
\OK &= \{x \in K : \lvert x \rvert \le 1\} \\
\mK &= \{x \in K : \lvert x \rvert < 1\}.
\end{align*}
Compute the residue fields of $\Q_p(i)$ and $\Q_p(\sqrt{p})$.
\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.

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