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\begin{document}
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\large
\textbf{18.785 Number Theory\hspace{228pt}Fall~2015}\\\vspace{4pt}
\textbf{Problem Set \#6\hspace{220pt}Due: 10/28/2015}
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\subsection*{Description}

These problems are related to the material covered in Lectures 10-12.
Your solutions are to be written up in latex (you can use the latex source for the problem set as a template) and submitted as a pdf-file with a filename of the form \texttt{SurnamePset6.pdf} via e-mail to \texttt{drew@math.mit.edu} by {\bf 5pm} on the date due.
Collaboration is permitted/encouraged, but you must identify your collaborators, and any references you consulted.
If there are none, write ``\textbf{Sources consulted:\ none}" at the top of your problem set.
The first person to spot each non-trivial typo/error in any of the problem sets or lecture notes will receive 1-5 points of extra credit.
\medskip

\noindent
\textbf{Instructions:} First do the warm up problems, then pick two of problems 1-3 to solve and write up your answers in latex.  Finally, be sure to complete the survey problem 4.

\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 the absolute discriminant of a number field must be a square mod $4$.
\item[(b)] Compute the different ideal of the quadratic extensions $\Q(\sqrt{-2})/\Q$ and $\Q(\sqrt{-3})/\Q$.
\item[(c)] Determine all the primes that ramify in the cubic fields $\Q[x]/(x^3-x-1)$ and $\Q[x]/(x^3+x+1)$ and compute their ramification indices.
\item[(d)] Let $p$ be an odd prime.  Compute the different ideal and absolute discriminant of the cyclotomic extension $\Q(\zeta_p)/\Q$.
\end{enumerate}

\subsection*{Problem 1 The different ideal (50 points)}

Let $A$ be a Dedekind domain with fraction field $K$, let $L/K$ be a finite separable extension, and let $B$ be the integral closure of $A$ in $L$.
Write $L=K(\alpha)$ with $\alpha\in B$ and let $f\in A[x]$ be the minimal polynomial of $\alpha$, with degree $n=[L:K]$.
\begin{itemize}
\item[(\bf a)] By comparing the Laurent series expansion of $1/f(x)$ with it partial fraction decomposition over the splitting field of $f$ (Galois closure of $L$), prove that
\[
\T_{L/K}\left(\frac{\alpha^i}{f'(\alpha)}\right)=\begin{cases}
0&\text{if }0\le i\le n-2;\\
1&\text{if }i=n-1;\\
\in A&\text{if }i\ge n.
\end{cases}
\]

\item[(\bf b)] Suppose $B=A[\alpha]$.  Prove that $B^*:=\{x\in L:T_{L/K}(xb)\in A\text{ for all }b\in B\}$ is the principal fractional $B$-ideal $(1/f'(\alpha))$.  Conclude that $\D_{B/A}=(f'(\alpha))$.

\item[(\bf c)] For any $\beta\in B$ with minimal polynomial $g\in A[x]$ define
\[
\delta_{B/A}(\beta)=\begin{cases}
g'(\beta) & \text{if } L=K(\beta);\\
0 & \text{otherwise}.
\end{cases}
\]
One can show that $\D_{B/A}$ is the $B$-ideal generated by $\{\delta_{B/A}(\beta):\beta\in B\}$ (you are not required to do this).
Prove that if $g$ is the minimal polynomial of $\beta\in B$ for which $L=K(\beta)$ then $N_{B/A}(g'(\beta))=\pm \disc(g)$.
Does this imply that $D_{B/A}$ is generated by the discriminants of all such $g$?  (you may wish to consider Dedekind's cubic field $L=\Q[x]/(x^3+x^2-2x+8)$ when answering this question).

\item[(\bf d)] Let $\c$ be the conductor of the order $C=A[\alpha]$.
Prove that
\[
\c=(B^*:C^*):=\{x\in L:xC^*\subseteq B^*\}.
\]
Conclude that if we define $\D_{C/A}:=(B:C^*)$ and $D_{C/A}=D(C)$ then we have $\D_{C/A}=\c\,\D_{B/A}$ and $D_{C/A}=\N_{B/A}(\c)D_{B/A}$, and that $D_{C/A}=\N_{B/A}(\D_{C/A})$.

\item[(\bf e)] Let $q$ be a prime of $B$ lying above a prime $p$ of $A$ and suppose the corresponding residue field extension is separable.  Prove that $v_\q(\D_{B/A})\ge e_\q-1$ with equality if and only if $B/A$ is tamely ramified at $\q$ (hint: reduce to the case where $A$ is a complete DVR and $B/A$ is totally ramified).

\item[(\bf f)] Let $p$ and $q$ be distinct primes congruent to $1\bmod 4$, let $K:=\Q(\sqrt{pq})$, and let $L:=\Q(\sqrt{p},\sqrt{q})$.  Prove that $\D_{L/K}$ is the unit ideal; thus $L/K$ is unramified.
\end{itemize}


\subsection*{Problem 2. Valuation rings (50 points)}
An \emph{ordered abelian group} is an abelian group $\Gamma$ with a total order $\le$ that is compatible with the group operation.
This means that for all $a,b,c\in \Gamma$ the following hold:
\[
\begin{matrix}
a\le b\le a &\implies &a=b &\text{(antisymmetry)}\\
a\le b\le c &\implies & a\le c &\text{(transitivity)}\\
a\not\le b &\implies & b\le a &\text{(totality)}\\
a\le b &\implies &a+c\le b+c &\text{(compatibility)}
\end{matrix}
\]
Note that totality implies reflexivity $(a\le a)$ .  Given an ordered abelian group $\Gamma$, we define the relations $\ge,<,>$ and the sets $\Gamma_{\le 0}, \Gamma_{\ge 0}, \Gamma_{<0}$, and $\Gamma_{>0}$ in the obvious way.

A \emph{valuation} $v$ on a field $K$ is a surjective homomorphism $v\colon K^\times\to \Gamma$ to an ordered abelian group $\Gamma$ that satisfies $v(x+y)\ge \min (v(x),v(y))$ for all $x,y\in K^\times$.
The group $\Gamma$ is called the \emph{value group} of $v$, and when $\Gamma=\{0\}$ we say that $v$ is the \emph{trivial valuation}.
We may extend $v$ to $K$ by defining $v(0)=\infty$, where $\infty$ is defined to be strictly greater than any element of $\Gamma$.

Recall that a \emph{valuation ring} is an integral domain $A$ with fraction field $K$ such that for all $x\in K^\times$ either $x\in A$ or $x^{-1}\in A$ (possibly both).

\begin{enumerate}
\item[({\bf a})]
Let $A$ be a valuation ring with fraction field $K$, and let $v\colon K^\times\to K^\times/A^\times= \Gamma$ be the quotient map.
Show that the relation $\le$ on $\Gamma$ defined by
\[
v(x)\le v(y) \Longleftrightarrow y/x\in A,
\]
makes $\Gamma$ an ordered abelian group and that $v$ is a valuation on $K$.

\item[({\bf b})]
Let $K$ be a field with a non-trivial valuation $v\colon K^\times\to \Gamma$.  Prove that
the set
\[
A:=\{x\in K:v(x)\ge 0\}
\]
is a valuation ring with fraction field $K$ and that $v(x)\le v(y) \Longleftrightarrow y/x\in A$.

\item[({\bf c})]
Let $\Gamma$ be an ordered abelian group and let $k$ be a field.
For each $a\in \Gamma_{\ge 0}$, let $x^a$ be a formal symbol, and define multiplication of these symbols via $x^ax^b:=x^{a+b}$.
Let~$A$ be the $k$-algebra whose elements are formal sums $\sum_{a\in I} c_a x^a$, where $c_a\in k$ and the index set $I\subseteq \Gamma_{\ge 0}$ is \emph{well-ordered} (every subset has a minimal element).
Let $K$ be the fraction field of $A$ and define $v\colon K^\times\to \Gamma$ by
\[
v\left(\frac{\sum c_a x^a}{\sum d_a x^a}\right) = \min\{ a: c_a\ne 0\} - \min\{ a:d_a\ne 0\}.
\]
Prove that $v$ is a valuation on $K$ with value group $\Gamma$ and valuation ring $A$.

\item[({\bf d})]
Let $v\colon K^\times\to \Gamma_v$ and $w\colon K^\times\to \Gamma_w$ be two valuations on a field $K$, and let $A_v$ and $A_w$ be the corresponding valuation rings.
Prove that $A_v=A_w$ if and only if there is an order preserving isomorphism $\rho\colon \Gamma_v\to \Gamma_w$ for which $\rho\circ v = w$, in which case we say that $v$ and $w$ are \emph{equivalent}.
Thus there is a 1-to-1 correspondence between valuation rings with fraction field $K$ and equivalence classes of valuations on $K$

\item[({\bf e})]
Let $A$ be an integral domain properly contained in its fraction field $K$, and let~$\mathcal{R}$ be the set of local rings that contain $A$ and are properly contained in $K$.
Partially order $\mathcal{R}$ by writing $R_1\le R_2$ if $R_1\subseteq R_2$ and the maximal ideal of $R_1$ is contained in the maximal ideal of $R_2$ (this is known as the \emph{dominance ordering}).
Prove that~$\mathcal{R}$ contains a maximal element $R$ and that every such $R$ is a valuation ring.

\item[({\bf f})] Prove that every valuation ring is local and integrally closed, and that the intersection of all valuation rings that contain an integral domain $A$ and lie in its fraction field is equal to the integral closure of $A$.

\item[({\bf g})] Prove that a valuation ring that is not a field is a discrete valuation ring if and only if it is noetherian.
\end{enumerate}

\subsection*{Problem 3. Minkowski's lemma and sums of four squares (50 points)}

Recall Minkowski's lemma (for $\Z^n$): \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 use this to prove Minkowski's lemma.

\item[(\bf b)] Prove that Minkowski's lemma is tight in the following sense: show that is 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 integers 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 the set of positive integers that are sums of four squares is closed under multiplication.
\item[(\bf h)] Prove that every positive integer is the sum of four squares.
\end{itemize}



\subsection*{Problem 4. Survey}
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
\end{tabular}
\end{center}
\noindent
Please rate each of the following lectures that you attended, according to the quality of the material (1=``useless", 10=``fascinating"), the quality of the presentation (1=``epic fail", 10=``perfection"), the pace (1=``way too slow", 10=``way too fast", 5=``just right") and the novelty of the material to you (1=``old hat", 10=``all new").

\begin{center}
\begin{tabular}{l|l|r|r|r|r|r}
Date & Lecture Topic & Material & Presentation & Pace & Novelty\\\hline
10/22 & Haar measure, product formula & & & & \\\hline 
10/27 & Minkowski bound, finiteness results  & & & & \\\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.


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