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\begin{document}
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\large
\textbf{18.785 Number Theory\hspace{228pt}Fall~2025}\\\vspace{4pt}
\textbf{Problem Set \#8\due{11/12/2025}}\\\vspace{-6pt}
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\subsection*{Description}

These problems are related to Lectures 12--15.
Your solutions should be written up in latex and submitted as a pdf-file \ifOCW\else to \href{https://www.gradescope.com/courses/1110057}{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)] Prove that a cubic field $K$ is Galois if and only if $D_K$ is a perfect square.
\item[(b)] Prove that our two definitions of a lattice $\Lambda$ in $V\simeq \R^n$ are equivalent: $\Lambda$ is a $\Z$-submodule generated by an $\R$-basis for $V$ if and only if it is a discrete cocompact subgroup of $V$.
\item[(c)] Let $n\in\Z_{>0}$ and assume $n^2-1$ is squarefree.  Prove that $n+\sqrt{n^2-1}$ is the fundamental unit of $\Q(\sqrt{n^2-1})$.
\end{enumerate}

\subsection*{Problem 1. Classification of global fields (66 points)}
Let $K$ be a field and let $M_K$ be the set of places of $K$ (equivalence classes of nontrivial absolute values).
We say that $K$ has a (strong) \emph{product formula} if $M_K$ is nonempty for each $v\in M_K$ there is an absolute value $|\ |_v$ in its equivalence class and a positive real number $m_v$ such that for all $x\in K^\times$ we have
\[
\prod_{v\in M_K}|x|_v^{m_v}=1,
\]
where all but finitely many factors in the product are equal to $1$.
Equivalently, if we fix \emph{normalized absolute values} $\|\ \|_v:=|x|_v^{m_v}$ for each $v\in M_K$, then for all $x\in K^\times$ we have
\[
\prod_{v\in M_K}\|x\|_v=1,
\]
with $\|x\|_v=1$ for all but finitely many $v\in M_K$.

\begin{definition}
A field $K$ is a \emph{global field} if it has a product formula and the completion $K_v$ of~$K$ at each place $v\in M_K$ is a local field.
\end{definition}

In Lectures 10 and 13 we proved every finite extension of $\Q$ and $\Fq(t)$ is a global field.  In this problem you will prove the converse, a result due to Artin and Whaples \cite{AW45}.
\smallskip

Let $K$ be a global field with normalized absolute values $\|\ \|_v$ for $v\in M_K$ that satisfy the product formula.
As we defined in lecture, an $M_K$-\emph{divisor} is a sequence of positive real numbers $c=(c_v)$ indexed by $v\in M_K$ with all but finitely many $c_v=1$ such that for each $v\in M_K$ there is an $x\in K_v^\times$ for which $c_v=\|x\|_v$.
For each $M_K$-divisor $c$ we define the set
\[
L(c):=\{x\in K:\|x\|_v\le c_v\text{ for all } v\in M_K\}.
\]

\begin{enumerate}
\item[(\bf a)] Let $E/F$ be a finite Galois extension. Prove $E$ is a global field if and only if $F$~is.
\item[(\bf b)] Extend your proof of (a) to all finite extensions $E/F$.
\item[(\bf c)] Prove that $M_K$ is infinite but contains only finitely many archimedean places.
\item[(\bf d)] Assume $K$ has an archimedean place.  Prove that $L(c)$ is finite for every $M_K$-divisor~$c$ (we proved this in class for number fields, but here $K$ is a global field as defined above).
\item[(\bf e)] Extend your proof of (d) to the case where $K$ has no archimedean places.
\item[(\bf f)] Prove that if $M_K$ contains an archimedean place then $K$ is a finite extension of~$\Q$ (hint: show $\Q\subseteq K$ and use (d) to show that $K/\Q$ is a finite extension).
\item[(\bf g)] Prove that if $M_K$ does not contain an archimedean place then $K$ is a finite extension of $\Fq(t)$ for some finite field $\Fq$ (hint: by choosing an appropriate $M_K$-divisor $c$, show that $L(c)$ is a finite field $k\subseteq K$ and that every $t\in K-k$ is transcendental over $k$; then show that $K$ is a finite extension of $k(t)$).
\item[(\bf h)] In your proofs of (a)-(g) above, where did you use the fact that the completions of~$K$ are local fields?
Show that if $K$ has a product formula and $K_v$ is a local field for any place $v\in M_K$ then $K_v$ is a local field for every place $v\in M_K$ (so we could weaken our definition of a global field to only require one $K_v$ to be a local field).
Are there fields with a product formula for which no completion is a local field?
\end{enumerate}

\subsection*{Problem 2. Finiteness of global class groups (66 points)}

A commutative ring $R$ with finite quotients by all nonzero ideals is a \emph{finite quotient domain}; to rule out trivial cases we further assume $R$ is not a field.  For such $R$ we define the \emph{absolute ideal norm} $\N_R(I):= \#R/I$ for each nonzero $R$-ideal $I$, let $\N_R((0)):=0$, and put  $\N_R(r):=\N_R((r))$ for $r\in R$.
\bigskip

Recall our standard $AKLB$ assumption: $A$ is a Dedekind domain, $K$ is its fraction field, $L/K$ is finite separable, and $B$ is the integral closure of $A$ in $L$.

\begin{enumerate}
\item[{\bf(a)}] Assume $AKLB$.  Show that if $A$ is a finite quotient domain then so is $B$, and we have the identity $\N_B(\alpha)=\N_A(\N_{L/K}(\alpha))$ for all $\alpha\in B$.
\end{enumerate}


\begin{definition}
A PID is \emph{basic} if it is a finite quotient domain and $\exists c_1,c_2\in \Z_{>0}$ such that
\begin{itemize}
\item[(i)] $\#\{x\in A:\N_A(x)\le c_1m\} \ge m$ for all integers $m$,
\item [(ii)] $\N_A(x+y)\le c_2(\N_A(x)+\N_A(y))$ for all $x,y\in A$.
\end{itemize}
\end{definition}

\begin{enumerate}
\item[{\bf(b)}] Show that $\Z$ is a basic PID but $\Z[\sqrt{5}]$ is not.  Is $\Z[i]$ a basic PID?
\item[{\bf(c)}] Show that $\Fq[t]$ is a basic PID.
\end{enumerate}

\begin{definition}
A \emph{global Dedekind domain} (over $A$) is a Dedekind domain that is free of finite rank as a module over a subring $A$ that is a basic PID.
\end{definition}

\begin{enumerate}
\item[{\bf(d)}] Assume $AKLB$.  Show that if $A$ is a basic PID then $B$ is a global Dedekind domain.  Conclude that every global field is the fraction field of a global Dedekind domain.
\item[{\bf(e)}] Let $B$ be a global Dedekind domain over a basic PID $A$ and let $e_1,\ldots e_n$ be an $A$-basis for $B$.  Prove there exists a homogeneous $f\in A[x_1,\ldots,x_n]$ of degree $n$ such that for any $\alpha\in B$,
\[
\N_{L/K}(\alpha) = f(a_1,\ldots,a_n),
\]
where $\alpha=a_1e_1+\cdots a_ne_n$ with $a_i\in A$.  Prove there exists $c\in \Z_{>0}$ such that
\[
\N_B(\alpha)\le c\max(\N_A(a_1),\ldots,\N_A(a_n))^n
\]
for all $\alpha=a_1e_1+\cdots a_ne_n\in B$.
\item[{\bf(f)}] Let $B$ be a global Dedekind domain. Prove that there is a real number $m$ such that for every nonzero $B$-ideal $I$ there is a nonzero $\alpha\in I$ for which $\N_B(\alpha)\le  m\N_B(I)$.

\item[{\bf(g)}] Prove that the ideal class group of a global Dedekind domain over $\Z$ or $\Fq(t)$ is finite (in fact all global Dedekind domains are global Dedekind domains over one of these basic PIDs, but you are not asked to prove this).

\end{enumerate}

\begin{remark}
Given (d) and the terminology we have chosen, you might wonder if the fraction field of a global Dedekind domain is necessarily a global field.  The  answer is yes!  You can find a proof of this in \cite[\S 4]{stasinski}, but I urge you to wait until you have solved this problem before reading it (this problem is adapted from \cite[\S 2-3]{stasinski} and \cite[\S 3]{swan}).
\end{remark}

\subsection*{Problem 3. Some applications of the Minkowski bound (33 points)}
For a number field $K$, let
\[
m_K:=\frac{n!}{n^n}\left(\frac{4}{\pi}\right)^s\sqrt{|D_K|}
\]
denote the Minkowski constant and let $h_K:=\#\cl\O_K$ denote the class number.
You may wish to use a computer to help with some of the calculations involved in this problem, but if you do so, please describe your computations (preferably in words or pseudo-code).

\begin{enumerate}[{\bf(a)}]
\item Prove that if $\O_K$ contains no prime ideals $\p$ of norm $\N(\p)\le m_K$ other than inert primes, then $h_K=1$, and show that when $K$ is an imaginary quadratic field the converse also holds.
\item Let $K$ be an imaginary quadratic field.  Show that if $h_K=1$ then $|D_K|$ is a power of $2$ or a prime congruent to $3\bmod 4$, and then determine all imaginary quadratic fields~$K$ of class number one with $|D_K|<200$ (this is in fact all of them).
\item Prove that there are no totally real cubic fields of discriminant less than 20 and that every totally real cubic field $K$ with $D_K<M$ can be written as $K=\Q(\alpha)$, where $\alpha$ is an algebraic integer with minimal polynomial $x^3+ax^2+bx+c$ whose coefficients satisfy $|a|<\sqrt{M}+2$, $|b|<2\sqrt{M}+1$, and $|c|<\sqrt{M}$.
\item Determine all totally real cubic fields $K$ that are ramified only at a prime $p<10$ and give a defining polynomial for each field that arises.  You may find the Sage function \texttt{pari.polredabs} useful: given a monic irreducible polynomial in $\Z[x]$ it will output another monic irreducible polynomial that defines the same number field but may have smaller discriminant (it will never be larger).

\end{enumerate}

\subsection*{Problem 4. Minkowski's lemma and sums of four squares (33 points)}

Minkowski's lemma (for $\Z^n$) states that \emph{if $S\subseteq \R^n$ is a symmetric convex set of volume $\mu(S)>2^n$ then $S$ contains a nonzero element of $\Z^n$}.

\medskip
Here \emph{symmetric} means that $S$ is closed under negation, and \emph{convex} means that for all $x,y\in S$ the set $\{tx+(1-t)y:t\in [0,1]\}$ lies in $S$.

\begin{itemize}
\item[{\bf (a)}] Prove that for any measurable $S\subseteq \R^n$ with measure $\mu(S)>1$ there exist distinct $s,t\in S$ such that $s-t\in \Z^n$, then prove Minkowski's lemma.

\item[{\bf (b)}] Prove that Minkowski's lemma is tight in the following sense: show that it is false if either of the words ``symmetric" or ``convex" is removed, or if the strict inequality $\mu(S)>2^n$ is weakened to $\mu(S)\ge 2^n$ (give three explicit counter examples).

\item[{\bf (c)}] Prove that one can weaken the inequality $\mu(S)>2^n$ in Minkowski's lemma to $\mu(S)\ge 2^n$ if $S$ is assumed to be compact.

\end{itemize}

You will now use Minkowski's lemma to prove a theorem of Lagrange, which states that every positive integer is a sum of four integer squares.
Let $p$ be an odd prime.

\begin{itemize}
\item[{\bf (d)}] Show that $x^2+y^2=a$ has a solution $(m,n)$ in $\Fp^2$ for every $a\in \Fp$.
\item[{\bf (e)}] Let $V$ be the $\Fp$-span of $\{(m,n,1,0), (-n,m,0,1)\}$ in $\Fp^4$, where $m^2+n^2=-1$.
Prove that $V$ is \emph{isotropic}, meaning that $v_1^2+v_2^2+v_3^2+v_4^2=0$ for all $v\in V$.
\item[{\bf (f)}] Use Minkowski's lemma to prove that $p$ is a sum of four squares.
\item[{\bf (g)}] Prove that every positive integer is the sum of four squares.
\end{itemize}


\subsection*{Problem 5. Unit groups of real quadratic fields (66 points)}

A (simple) \emph{continued fraction} is a (possibly infinite) expression of the form
\[
a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cdots}}
\]
with $a_n\in \Z$ and $a_n>0$ for $n>0$.
They are more compactly written as $(a_0;a_1,a_2,\ldots)$.
For any $t\in \R_{> 0}$ the \emph{continued fraction expansion} of $t$ is defined recursively via
\[
t_0:=t,\qquad a_n :=\lfloor t_n\rfloor,\qquad  t_{n+1}:=1/(t_n-a_n),
\]
where the sequence $a(t):=(a_0;a_1,a_2,\ldots)$ terminates at $a_n$ if $t_n=a_n$, in which case we say that $a(t)=(a_0;a_1,\ldots,a_n)$ is \emph{finite}, and otherwise call $a(t)=(a_0;a_1,a_2,\ldots)$ \emph{infinite}.
If $a(t)$ is infinite and there exists $\ell\in \Z_{>0}$ such that $a_{n+\ell}=a_n$ for all sufficiently large $n$,
we say that $a(t)$ is \emph{periodic} and call the least such integer $\ell:=\ell(t)$ the \emph{period} of $a(t)$.

For an infinite continued fraction $a(t):=(a_0;a_1,a_2,\ldots)$, we define $P_n,Q_n\in \Z_{\ge 0}$ via
\begin{align*}
P_{-2}&=0,\qquad\, P_{-1}=1,\qquad\, P_n=a_nP_{n-1}+P_{n-2};\\
Q_{-2}&=1,\qquad Q_{-1}=0,\qquad Q_n=a_nQ_{n-1}+Q_{n-2}.
\end{align*}
Note that $P_{n+2}>P_{n+1}\ge P_n>0$ and $Q_{n+1}> Q_n\ge Q_{n-1} > 0$ for all $n\ge 1$.

\begin{enumerate}[(\bf a)]
\item Prove that $a(t)$ is finite if and only if $t\in\Q$, in which case $t=a(t)$.
\item Prove that if $a(t)$ is infinite then for all $n\ge 0$ we have
\[
P_{n-2}Q_{n-1}-P_{n-1}Q_{n-2}=\pm 1\qquad\text{and}\qquad t = \frac{t_nP_{n-1}+P_{n-2}}{t_nQ_{n-1}+Q_{n-2}}.
\]
\item Prove that if $a(t)$ is infinite then $P_n/Q_n=(a_0;a_1,\ldots,a_n)$ with $|Q_n^2t-P_nQ_n|<1$ and $a(t_n)=(a_n;a_{n+1},a_{n+2},\ldots)$, for all $n\ge 0$. Conclude that $t=\lim_{n\to\infty} P_n/Q_n = a(t)$.
\item  Prove that if $a(t)$ is infinite then for any $P,Q\in \Z_{>0}$ with $|tQ-P| < 1/(2Q)$ we have $P/Q=P_n/Q_n$ for some $n\ge 0$.

\item Prove that $a(t)$ is periodic if and only if $t$ is a real quadratic irrational.\\
(Hint: show that if $At^2+Bt+C=0$ with $A,B,C\in \Z$ then $A_n t_n^2+B_nt_n+C_n=0$ with $A_n,B_n,C_n\in \Z$ and $B_n^2-4A_nC_n=B^2-4AC$ and $|A_n|,|C_n|\le 2|At|+|A|+|B|$).
\end{enumerate}

\begin{definition}
Let $t$ be a real quadratic irrational with Galois conjugate $t'$ and continued fraction $a(t)=(a_0;a_1,a_2,\ldots)$ of period $\ell=\ell(t)$.
We say that $t$ is \emph{purely periodic} if we have $a_i=a_{\ell+i}$ for all $i\ge 0$ and call $t$ \emph{reduced} if $t>1$ and $-1<t'<0$,
\end{definition}

\begin{enumerate}[(\bf a)]
\setcounter{enumi}{5}
\item Prove that a real quadratic irrational $t$ is reduced if and only if it is purely periodic.
\end{enumerate}

Now let $D>0$ be a squarefree integer that is not congruent to $1\bmod 4$ and let $K=\Q(\sqrt{D})$.
As shown on previous problem sets, $\O_K=\Z[\sqrt{D}]$, and it is clear that $(\O_K^\times)_{\rm tors}=\{\pm 1\}$.
Every $\alpha=x+y\sqrt{D}\in\O_K^\times$ has $N(\alpha)=\pm 1$, and $(x,y)$ is thus an (integer) solution to the \emph{Pell equation}
\begin{equation}\label{eq:pell}
X^2-DY^2=\pm 1
\end{equation}
\begin{enumerate}[(\bf a)]
\setcounter{enumi}{6}
\item Prove that if $(x_1,y_1)$ and $(x_2,y_2)$ are solutions to \eqref{eq:pell} with $x_1,y_1,x_2,y_2\in\Z_{>0}$ then $x_1+y_1\sqrt{D} < x_2+y_2\sqrt{D}$ if and only if $x_1<x_2$ and $y_1\le y_2$.  Conclude that the fundamental unit $\epsilon=x+y\sqrt{D}$ of $\O_K^\times$ is the unique solution $(x,y)$ to \eqref{eq:pell} with $x,y>0$ and $x$ minimal.
\item Let $a(\sqrt{D})=(a_0;a_1,a_2,\ldots)$, define $t_n,P_n,Q_n$ as above, and let $\ell:=\ell(\sqrt{D})$.
Prove that $(P_{k\ell-1},Q_{k\ell-1})$ is a solution to \eqref{eq:pell} for all $k\ge 0$ and that $\epsilon=P_{\ell-1}+Q_{\ell-1}\sqrt{D}$.
\item Compute the fundamental unit $\epsilon$ for each of the real quadratic fields $\Q(\sqrt{19})$, $\Q(\sqrt{570})$, and $\Q(\sqrt{571})$; in each case give the period $\ell(\sqrt{D})$ as well as $\epsilon$.
\end{enumerate}


\subsection*{Problem 6. $S$-class groups and $S$-unit groups (33 points)}
Let $K$ be a number field with ring of integers $\O_K$, and let $S$ be a finite set of places of~$K$ including all archimedean places.
Define the \emph{ring of $S$-integers} $\O_{K,S}$ as the set
\[
\O_{K,S}:=\{x\in K:v_\p(x)\ge 0\text{ for all }\p\not\in S\}.
\]
\begin{enumerate}
\item[(\bf a)] Prove that $\O_{K,S}$ is a Dedekind domain containing $\O_K$ with the same fraction field.
\item[(\bf b)] Define a natural homomorphism between $\cl\O_{K,S}$ and $\cl\O_K$ (it is up to you to determine which direction it should go) and use it to prove that $\cl \O_{K,S}$ is finite.
\item[(\bf c)] Prove that there is a finite set $S$ for which $\O_{K,S}$ is a PID and give an explicit upper bound on $\#S$ that depends only on $n=[K:\Q]$ and $|\disc \O_K|$.
\item[(\bf d)] Prove the \emph{$S$-unit theorem}: $\O_{K,S}^\times$ is a finitely generated abelian group of rank $\#S-1$.
\end{enumerate}


\subsection*{Problem 7. 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
Problem 6 & & & \\\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}:

\begin{thebibliography}{99}
\bibitem{AW45}
Emil Artin and George Whaples, \href{http://www.ams.org/journals/bull/1945-51-07/S0002-9904-1945-08383-9/S0002-9904-1945-08383-9.pdf}{\textit{Axiomatic characterization of fields by the product formula for valuations}}, Bull. Amer. Math. Soc. \textbf{51} (1945), 469--492.
\bibitem{stasinski}
Alexander Stasinski, \href{https://doi.org/10.1080/00029890.2021.1855036}{\textit{A uniform proof of the finiteness of the class group of a global field}}, American Mathematical Monthly \textbf{1228} (2021),  239--249.
\bibitem{swan}
Richard G. Swan and E. Graham Evans, \href{https://link.springer.com/book/10.1007/BFb0059150}{\textit{$K$-theory of finite groups and orders}}, Lecture Notes in Mathematics \textbf{149}, Springer, 1970.
\end{thebibliography}

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