\newif\ifOCW
%\OCWtrue
\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{hyperref}
\hypersetup{colorlinks=true,urlcolor=blue,citecolor=blue,linkcolor=blue}
\usepackage{courier}
\usepackage{tikz}
\usetikzlibrary{calc,matrix,arrows,decorations.markings}
\usepackage{array}
\usepackage{color}
\usepackage{enumerate}
\usepackage{nicefrac}
\usepackage{listings}
\ifOCW\usepackage{soul}\let\oldhref\href\renewcommand{\href}[2]{\oldhref{#1}{\ul{#2}}}\fi
\ifOCW\newcommand{\due}[1]{\hfill\phantom{Due: #1}}\else\newcommand{\due}[1]{\hfill{Due: #1}}\fi
\lstset{
	basicstyle=\small\ttfamily,
	keywordstyle=\color{blue},
	language=python,
	xleftmargin=16pt,
}

\textwidth=5.8in
\textheight=9in
\topmargin=-0.5in
\headheight=0in
\headsep=.5in
\hoffset  -.4in
\pagestyle{plain}

% growing list of useful macros, use these where appropriate and add to this list as needed
\newcommand{\kbar}{\bar{k}}
\newcommand{\Fp}{\mathbb{F}_p}
\newcommand{\Fpbar}{\overline{\mathbb{F}}_p}
\newcommand{\Fq}{\mathbb{F}_q}
\newcommand{\Fqbar}{\overline{\mathbb{F}}_q}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Qbar}{\overline{\mathbb{Q}}}
\newcommand{\Qp}{\Q_p}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Zp}{\Z_p}
\newcommand{\Aut}{{\rm Aut}}
\newcommand{\Gal}{{\rm Gal}}
\newcommand{\GL}{{\rm GL}}
\newcommand{\PGL}{{\rm PGL}}
\newcommand{\dy}{\,dy}
\newcommand{\dx}{\,dx}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\kron}[2]{\bigl(\frac{#1}{#2}\bigr)}
\newcommand{\lcm}{\operatorname{lcm}}
\newcommand{\ceil}[1]{\lceil{#1}\rceil}
\newcommand{\Exp}{{\rm E}}
\renewcommand{\O}{\mathcal{O}}
\newcommand{\OK}{\O_K}
\newcommand{\T}{{\rm T}}
\newcommand{\N}{{\rm N}}
\newcommand{\re}{\operatorname{re}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\ord}{{\rm ord}}
\newcommand{\cl}{{\rm cl}}
\newcommand{\disc}{{\rm disc}}
\newcommand{\p}{\mathfrak{p}}
\newcommand{\q}{\mathfrak{q}}
\newcommand{\m}{{\mathfrak m}}
\renewcommand{\c}{{\mathfrak c}}
\renewcommand{\a}{{\mathfrak a}}
\newcommand{\Frob}{{\rm Frob}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\tor}{{\rm tors}}
\newcommand{\I}{\mathcal{I}}

\newtheorem*{theorem}{Theorem}

\begin{document}
\setlength{\unitlength}{1in}
\begin{center}
\large
\textbf{18.785 Number Theory\hspace{228pt}Fall~2025}\\\vspace{4pt}
\textbf{Problem Set \#5\due{10/13/2025}}\\\vspace{-6pt}
\normalsize
\begin{picture}(5.8,.1) 
\put(0,0) {\line(1,0){5.8}}
\end{picture}
\end{center}

\noindent
These problems are related to Lectures 6--8.
Your solutions should be written up in latex and submitted as a pdf-file \ifOCW\else to \href{https://www.gradescope.com/courses/283349}{Gradescope} \fi by midnight on the date due.
\medskip

\noindent
\textbf{Instructions:} Solve Problem 0, then pick any combination of problems that sum to 100 points. Collaboration is permitted/encouraged, but you must identify your collaborators (including any LLMs you consulted) and any references you consulted outside the course \ifOCW syllabus \else\href{https://math.mit.edu/classes/18.785/2025/syllabus.html}{syllabus}\fi.
Include this information after the \textbf{Collaborators/Sources} prompt at the end of the problem set  (if there are none, you should enter ``none'', do not leave it blank).
Note that each student is expected to write their own solutions; it is fine to discuss problems with others, but your writing must be your own.

\subsection*{Problem 0.}
These are warm-up problems that do not need to be turned in.
\begin{enumerate}
\setlength\itemsep{0pt}
\item[(a)] Give an example of a metric on a field that is not induced by an absolute value.
\item[(b)] Prove that the completion $\hat k$ of a field $k$ at one of its absolute values $|\ |$ satisfies the following universal property: every topological field embedding of $k$ into a complete field $k'$ extends uniquely to an embedding of $\hat k$ into $k'$ that is an isomorphism if and only if $k$ is dense in $k'$.
\item[(c)] Compute the $3$-adic expansions of $1/4$, $-5/6$ and $\sqrt{7}$ in $\Q_3$.
\item[(d)] Let $X$ be a metric space defined by a nonarchimedean absolute value.  Verify that (1) every point in an open ball is a center, (2) two open balls are either disjoint or concentric, (3) every open ball is closed and every closed ball is open,
(4) all triangles are isosceles, (5) $X$ is totally disconnected.
\item[(e)] Show that every $\alpha\in\Qp^\times$ can be written uniquely in the form $\alpha=p^ru$ for some $r\in\Z$ and $u\in\Zp^\times$.
\end{enumerate}

\subsection*{Problem 1. Quadratic reciprocity (33 points)}

Recall that for an odd prime $p$ the \emph{Legendre symbol} $\bigl(\frac{\cdot}{p}\bigr)\colon \Z\to \{-1,0,1\}$ is defined by
\[
\left(\frac{n}{p}\right):=\begin{cases}
-1 & \text{if $n$ is not a square modulo $p$};\\
0 & \text{if $n$ is divisible by $p$};\\
1 & \text{if $n$ is a nonzero square modulo $p$}.
\end{cases}
\]
Gauss's theorem of quadratic reciprocity states that for odd primes $p\ne q$:
\[
(1)\ \ \left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{(p-1)(q-1)}{4}};\quad
(2)\ \ \left(\frac{-1}{p}\right)=(-1)^{\frac{p-1}{2}};\quad
(3)\ \ \left(\frac{2}{p}\right)=(-1)^{\frac{p^2-1}{8}}.
\]
For any integer $n>1$, let $\zeta_n$ denote a primitive $n$th root of unity.

\begin{enumerate}
\item[\textbf{(a)}] Prove that $\Q(\zeta_n)/\Q$ is a Galois extension with  $\Gal(\Q(\zeta_n)/\Q)\simeq (\Z/n\Z)^\times$.
\item[\textbf{(b)}] Let $n>1$ be an integer, let $p$ be a prime that does not divide $n$ (this implies that $p$ is unramified in $\Q(\zeta_n)$, which you are not required to prove), and let $[p]$ denote the residue class of $p$ in $(\Z/n\Z)^\times\simeq\Gal(\Q(\zeta_n)/\Q)$.  Prove that
\[
\left(\frac{\Q(\zeta_n)/\Q}{(p)}\right) = [p],
\]
and conclude that for the extension $\Q(\zeta_n)/\Q$, the Artin map is surjective.
\item[\textbf{(c)}] Let $p$ be an odd prime, and define $p^*:=(-1)^{(p-1)/2}p$.
Prove that $\Q(\sqrt{p^*})$ is the unique quadratic subfield of $\Q(\zeta_p)$.
\item[\textbf{(d)}] By comparing values of the Artin map for suitably chosen cyclotomic and quadratic extensions of $\Q$, prove (1), (2), (3) (you may prove these in any order you wish).
\end{enumerate}

\subsection*{Problem 2. Weak approximation (33 points)}
Let $k$ be a field and for $n\in \Z_{\ge 1}$ let $S_n$ and $W_n$ denote the following statements:

\begin{itemize}
\item[$S_n:$] \emph{Given inequivalent nontrivial absolute values $|\ |_1,\ldots,|\ |_n$ on $k$, there is an $x\in k^\times$ for which $|x|_1>1$ and $|x|_i<1$ for $1<i\le n$.}
\item[$W_n:$] \emph{Given inequivalent nontrivial absolute values $|\ |_1,\ldots,|\ |_n$ on $k$, there is a sequence $(x_1,x_2,\ldots)$ of elements $x_j\in k$ that converges to $1$ with respect to $|\ |_1$ and to $0$ with respect to $|\ |_i$ for $1< i \le n$.}
\end{itemize}

\begin{enumerate}
\item[\textbf{(a)}] Prove that $S_n$ implies $W_n$.
\item[\textbf{(b)}] Prove that $S_n$ holds for all $n\ge 1$.
\item[\textbf{(c)}] Prove the Weak Approximation Theorem:

\emph{Given inequivalent nontrivial absolute values $|\ |_1,\ldots,|\ |_n$ on $k$, $a_1,\ldots,a_n\in k$, and $\epsilon_1,\ldots,\epsilon_n\in \R_{>0}$ there exists $x\in k$ such that $|x-a_i|_i<\epsilon_i$ for $i=1,\ldots,n$.}
\item[\textbf{(d)}] Let $|\ |_1$ and $|\ |_2$ be absolute values on $k$.  Prove that the topologies on $k$ induced by $|\ |_1$ and $|\ |_2$ coincide if and only if $|\ |_1\sim |\ |_2$.
\end{enumerate}

\subsection*{Problem 3. $n$-adic rings (66 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 nonarchimedean triangle inequality $|x+y|_n \le \max(|x|_n,|y|_n)$; in particular, $d_n(x,y):= |x-y|_n$ is a nonarchimedean metric.
\end{enumerate}
Let $A_k=\Z/n^k\Z$ and consider the inverse system consisting of the sequence 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 $\Z_n:=\varprojlim A_k$.
\begin{enumerate}
\item[\textbf{(b)}] Compute the first three terms of the $10$-adic expansions of $-7$, $1/3$, and $\sqrt[3]{3}$ in $\Z_{10}$ (as with the $p$-adic expansion defined in Lecture 8, each term is a decimal digit).
\item[\textbf{(c)}] For $n=p$ prime prove the fraction field of $\Z_p$ is (canonically isomorphic to)~$\Qp$, the completion of $\Q$ with respect to $|\ |_p$, and that $\Z_p$ is its valuation ring.

\item[\textbf{(d)}] Prove that $\Z_n$ is an integral domain if and only if $n$ is a prime power.
\end{enumerate}

In view of (d), we cannot construct the fraction field of $\Z_n$ in general,
but we can still define $\Q_n$ as the completion of $\Q$ with respect to the metric $d_n(x,y):=|x-y|_n$.

\begin{enumerate}
\item[\textbf{(e)}] Prove that $\Q_n$ is a ring containing (subrings canonically isomorphic to) $\Q$ and $\Z_n$.
\item[\textbf{(f)}] Extend $|\ |_n$ to $\Q_n$ and show that $d_n(x,y)=|x-y|_n$ for all $x,y\in \Q_n$.\\
Is $\Q_n$ a topological ring?
\item[\textbf{(g)}] For $n=p^e$ a prime power, prove that $\Q_n\simeq\Qp$ (as topological fields).
\item[\textbf{(h)}] Prove that in general we have a ring isomorphism $\Q_n\simeq \prod_{p|n}\Qp$.\\
If you answered yes to (f), do we also have an isomorphism of topological rings?
\end{enumerate}

\subsection*{Problem 4. Quadratic extensions of $\Qp$ (33 points)}

\begin{enumerate}
\item[\textbf{(a)}] Let $p\equiv 3\bmod 4$ be prime, and let $\p$ be the prime of $\Q(i)$ lying above $p$.
Let $\Qp(i)$ denote the extension of $\Qp$ obtained by adjoining a square root of $-1$, and let $\Q(i)_\p$ denote the completion of $\Q(i)$ at the absolute value $|\cdot|_\p$.
Show that $\Qp(i)$ has a unique absolute value extending the $p$-adic absolute value $|x|_p:= p^{-v_p(x)}$, and that $\Qp(i)$ and $\Q(i)_\p$ are isomorphic local fields.
Are their absolute values the same?
\item[\textbf{(b)}] Let $p$ be an odd prime.  Prove that $\Qp$ has exactly 3 distinct quadratic extensions; describe them explicitly, determine which are ramified, and compute their residue fields (the quotient of the ring of integers by its unique maximal ideal).
\item[\textbf{(c)}] Prove that $\Q_2$ has exactly 7 distinct quadratic extensions; describe them explicitly, determine which are ramified, and compute their residue fields.
\item[\textbf{(d)}] Prove that for every positive integer $n$ there exists a global number field (finite extension of $\Q$) with Galois group isomorphic to $(\Z/2\Z)^n$, but that for local number fields (finite extensions of $\Qp$ for some prime $p$) this occurs only for $n\le 3$.
\end{enumerate}

\subsection*{Problem 5. Roots of unity in $\Qp$ (33 points)}
Let $\Qp^{\times n}=\{x^n:x\in\Qp^\times\}$ denote the set of $n$th powers in $\Qp^\times$.

\begin{enumerate}
\item[\textbf{(a)}] Prove that $\Qp^\times/\Qp^{\times 2} \simeq (\Z/2\Z)^2$ when $p$ is odd, and $\Q_2^\times/\Q_2^{\times 2}\simeq (\Z/2\Z)^3$.\\
(hint: use Hensel's lemmas).
\item[\textbf{(b)}] Determine the structure of $\Qp^\times/\Qp^{\times n}$ for all primes $p$ and odd primes $n$.
\end{enumerate}
Let $\mu_{n,p}=\{x\in\Qp^\times:x^n=1\}$ denote the set of $n$th roots of unity in $\Qp$.
\begin{enumerate}
\item[\textbf{(c)}] Prove that $\mu_{n,p}$ is a cyclic group of order $\gcd(n,p-1)$ whenever $p\nmid n$, and that $\mu_{p,p}$ is trivial when $p$ is odd.
\item[\textbf{(d)}] Prove that the roots of unity in $\Qp$ form a cyclic subgroup of $\Zp^\times$ that has order $p-1$ when $p$ is odd, and order $2$ when $p=2$.
\end{enumerate}


\subsection*{Problem 6. Survey (1 point)}
Complete the following survey by rating each problem you attempted on a scale of 1 to~10 according to how interesting you found it (1 = ``mind-numbing," 10 = ``mind-blowing"), and how difficult you found it (1 = ``trivial," 10 = ``brutal").  Also estimate the amount of time you spent on each problem to the nearest half hour.

\begin{center}
\begin{tabular}{l|r|r|r|}
& Interest & Difficulty & Time Spent\\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
Problem 3 & & & \\\hline
Problem 4 & & & \\\hline
Problem 5 & & & \\\hline
\end{tabular}
\end{center}
\noindent
Please feel free to record any additional comments you have on the problem sets and the lectures, in particular, ways in which they might be improved.
\medskip

\noindent
\textbf{Collaborators/Sources}:

\end{document}