\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} \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{empty} % 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{\Fqm}{\mathbb{F}_{q^m}} \newcommand{\Fqn}{\mathbb{F}_{q^n}} \newcommand{\F}{\mathbb{F}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Qbar}{\overline{\Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Zhat}{\widehat\Z} \newcommand{\Zbar}{\overline\Z} \newcommand{\Kbar}{\overline K} \newcommand{\NN}{\mathbb{N}} \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}\frac{1}eil} \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}} \newcommand{\D}{\mathcal{D}} \newtheorem*{theorem}{Theorem} \theoremstyle{definition} \newtheorem*{definition}{Definition} \begin{document} \setlength{\unitlength}{1in} \begin{center} \large \textbf{18.785 Number Theory\hspace{228pt}Fall~2018}\\\vspace{4pt} \textbf{Problem Set \#6\hspace{220pt}Due: 10/24/2018} \normalsize \begin{picture}(5.8,.1) \put(0,0) {\line(1,0){5.8}} \end{picture} \end{center} \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 noon} 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 nontrivial 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 problem, then pick problems that sum to 96 points to solve and write up your answers in latex. Finally, complete the survey problem 5. \subsection*{Problem 0.} These are warm up questions that do not need to be turned in. \begin{enumerate} \setlength\itemsep{0pt} \item[(a)] Prove that the absolute discriminant of a number field is always 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 (64 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 its partial fraction decomposition over the splitting field of $f$ (the 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 prove this). Prove that if $g$ is the minimal polynomial of $\beta\in B$ for which $L=K(\beta)$ then $\N_{L/K}(g'(\beta))=\pm \disc(g)$. \item[(\bf d)] Prove or disprove: $D_{B/A}$ is the $A$-ideal generated by $\{\N_{L/K}(\delta_{B/A}(\beta)):\beta\in B\}$. \item[(\bf e)] 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 $D_{C/A}=\N_{B/A}(\c)D_{B/A}$, so that $D_{C/A}=\N_{B/A}(\D_{C/A})$. \item[(\bf f)] 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 \[ e_\q-1 \le v_\q(\D_{B/A})\le e_\q-1 + v_\q(e_\q), \] and that the lower bound is an equality only when $B/A$ is tamely ramified at $\q$. \item[(\bf g)] 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 (so $L/K$ is unramified). \end{itemize} \subsection*{Problem 2. Valuation rings (64 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 $\Gamma_{\le 0}, \Gamma_{\ge 0}, \Gamma_{<0}$, $\Gamma_{>0}$ 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).\footnote{That $A$ is a ring is a classical result of Hahn \cite{hahn}; see \cite[Thm.\ 5.1]{higman} for a modern proof.} 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. Norm maps of local fields (32 points)} Let $A$ be the valuation ring of a nonarchimedean local field $K$, let $L$ be a tamely ramified finite abelian extension of $K$, and let $B$ be the integral closure of $A$ in $L$. The goal of this problem is to prove that the extension $L/K$ is unramified if and only if the norm map restricts to a surjective map of unit groups, equivalently, $\N_{L/K}(B^\times)=A^\times$. Let $\p$ and $\q$ be the maximal ideals of $A$ and $B$ and $k:=A/\p$ and $l:=B/\q$ the residue fields. \begin{enumerate} \item[(\bf a)] Prove that we always have $\N_{L/K}(B^\times)\subseteq A^\times$ and $\N_{l/k}(l^\times)=k^\times$ and $\T_{l/k}(l)=k$. \item[(\bf b)] For $i\ge 0$ define $U_i:=1+\p^i:=\{1+a:a\in \p^i\}$. Show that the $U_i$ are distinct closed subgroups of $A^\times$ that form a base of neighborhoods $1\in A^\times$ (this means every open neighborhood of $1$ in the topological group $A^\times$ contains some $U_i$). \item[(\bf c)] Prove that if $L/K$ is totally ramified then the norm of every $b\in B^\times$ lies in a coset of $U_1$ of the form $u^nU_1$, where $n=[L:K]$. Show that for $n>1$ the norms of these cosets do not cover $A^\times$. Conclude that if $\N_{L/K}(B^\times)=A^\times$ then $L/K$ must be unramified. \end{enumerate} \begin{enumerate} \item[(\bf d)] Assume $L/K$ is unramified. Show that for every $u\in A^\times$ there exists $\alpha_0\in B^\times$ with $\N_{L/K}(\alpha_0)\equiv u\bmod \p$. Then construct $\alpha_1\in B^\times$ with $\N_{L/K}(\alpha_0\alpha_1)\equiv u \bmod \p^2$. Continuing in this fashion, construct $\alpha\in B^\times$ such that $\N_{L/K}(\alpha)=u$. \end{enumerate} \subsection*{Problem 4. Minkowski's lemma and sums of four squares (32 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 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 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. 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 Problem 4 & & & \\\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 10/17 & Different and discriminant ideals & & & & \\\hline 10/22 & Haar measure, product formula & & & & \\\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. \begin{thebibliography}{99} \bibitem{hahn} H. Hahn, \emph{\"Uber die nichtarchimedischen Gr\"ossensysteme}, Sitzungsberichte der K. Akademie der Wissenschaften, Vienna \textbf{116} (1907), 601--655. \bibitem{higman} G. Higman, \href{https://doi.org/10.1112/plms/s3-2.1.326}{\emph{Ordering by divisibility in abstract algebras}}, Proceedings of the London Mathematical Society (3) \textbf{2} (1952), 326--336. \end{thebibliography} \end{document}