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\begin{document}
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\large
\textbf{18.785 Number Theory\hspace{228pt}Fall~2019}\\\vspace{4pt}
\textbf{Problem Set \#2\hspace{220pt}Due: 09/18/2019}
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\subsection*{Description}

These problems are related to the material covered in Lectures 3--4.
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 named \texttt{SurnamePset2.pdf} (but replace \texttt{Surname} with your surname) via e-mail to \texttt{drew@math.mit.edu} by 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 typo/error in any of the problem sets or lecture notes will receive 1--5 points of extra credit.
\medskip

\noindent
\textbf{Instructions:} Solve Problem 0 (take the time to at least convince yourself that you know how to solve each part; come to office hours if you do not), then pick one of Problems 1-2 and one of Problems 3-4 to solve. Finally, complete the survey, Problem~5.

\subsection*{Problem 0. Warmup (0 points)}
These warmup exercises do not need to be written up.
I urge you to at least think through these problems (they should not take long).
\begin{enumerate}
\setlength\itemsep{0pt}
\item[\textbf{(a)}] Let $I,J,K$ be nonzero ideals in a noetherian (not necessarily Dedekind) domain.  Show that $(I:J+K)=(I:J)\cap(I:K)$.
\item[\textbf{(b)}] Let $K$ be the field $\Fp(x,y)$ and consider the field $L:=K[t]/(t^{p^2}+t^px+y)$.  Show that $L/K$ can be decomposed as a purely inseparable extension of a separable extension, but not as a separable extension of a purely inseparable extension.
\item[\textbf{(c)}] Let $K$ and $L$ be two number fields.  Describe the finite \`etale $K$-algebra $L\otimes_\Q K$ when $L\subseteq K$, $K\subseteq L$, $K=L$, $K\cap L=\Q$, and then in general.
\item[\textbf{(d)}] Let $K=\Q(\zeta_5)$ be the number field generated by a primitive 5th root of unity~$\zeta_5$.
Show that $K\otimes_\Q\R$ is isomorphic to $\R^4$ as an $\R$-vector space but not as an $\R$-algebra.
\item[\textbf{(e)}] Let $p$ be a prime.  Prove that there are exactly four (unital) commutative rings of cardinality $p^2$, two of which are finite \`etale $\Fp$-algebras.
Of these four, which arise as $A/I$ for some discrete valuation ring $A$ and ideal $I$?
\end{enumerate}


\subsection*{Problem 1. Characterizing Dedekind domains (64 points)}
Recall that we defined a Dedekind domain to be an integrally closed noetherian domain of dimension at most one, or equivalently, a noetherian domain whose localizations at nonzero prime ideals are discrete valuation rings (see Proposition 2.9); let (D) denote either of these equivalent conditions.
In Lecture~3 we proved that every Dedekind domain~$A$ enjoys the following properties:
\begin{enumerate}[(a)]
\item Each nonzero prime ideal of $A$ is invertible.
\item Each nonzero ideal of $A$ is a (finite) product of prime ideals.
\item $A$ is noetherian and \emph{to contain is to divide}: $J\supseteq I\Rightarrow J|I$ for all ideals $I,J$.
\item For each ideal $I$ in $A$ there exists a nonzero ideal $J$ such that $IJ$ is principal.
\item The quotient $A/I$ of $A$ by any nonzero ideal $I$ is a principal ideal ring.
\item If $a$ is a nonzero element of an ideal $I$ then $I=(a,b)$ for some $b\in I$.
\end{enumerate}

In this problem you will prove that for any integral domain $A$, each of the conditions above implies~(D).
You may prove these implications any order (e.g. it suffices to just prove (a)$\Rightarrow$(b) in your answer for part \textbf{(a)} so long as you eventually prove (b)$\Rightarrow$(D)).  You may want to first consider the case where $A$ is a noetherian local domain.

\begin{enumerate}
\item[\textbf{(a)}] Prove (a)$\Rightarrow$(D).
\item[\textbf{(b)}] Prove (b)$\Rightarrow$(D).
\item[\textbf{(c)}] Prove (c)$\Rightarrow$(D).
\item[\textbf{(d)}] Prove (d)$\Rightarrow$(D).
\item[\textbf{(e)}] Prove (e)$\Rightarrow$(D).
\item[\textbf{(f)}] Prove (f)$\Rightarrow$(D).
\item[\textbf{(g)}] Show that the noetherian integral domain $A=\Z[\sqrt{-3}]$ of dimension one is \emph{not} a Dedekind domain in two ways: show that it is not integrally closed and exhibit a nonzero prime ideal $\p$ for which $A_\p$ is not a DVR.  Then give similarly explicit demonstrations that $A$ does not satisfy each of the properties (a)-(f) above.
\end{enumerate}

\subsection*{Problem 2. Fermat's last theorem (64 points)\footnote{This problem is adapted from \cite[I, Ex.17-27]{lorenzini} but corrects/clarifies a number of minor issues there.}}
Recall that Fermat's Last Theorem (FLT) states that
\[
x^n+y^n=z^n
\]
has no integer solutions with $xyz\ne 0$ for $n>2$.
By removing common factors we may assume $\gcd(x,y,z)=1$, and we may assume that $n$ is a prime $p\ge 5$, since the cases $n=3$ and $n=4$ were proved by Euler and Fermat (respectively), and we can easily reduce to the case where either $n=p$ is prime or $n=4$ (every solution with $n=ab$ also gives a solution with $n=a$ and $n=b$).

So let $p\ge 5$ be prime and suppose $x,y,z$ are relatively prime integers for which
\[
x^p+y^p=z^p
\]
with $xyz\ne 0$, and let $\zeta_p\in \Qbar$ denote a primitive $p$th root of unity (so $\zeta_p^p=1$ but $\zeta_p\ne 1$).
In order to simplify matters, we will make two further assumptions:
\begin{enumerate}
\item[(1)] $xyz\ne 0\bmod p$;
\item[(2)] the ring $\Z[\zeta_p]$ is a UFD.
\end{enumerate}
You will prove below that under these assumptions, no such $x,y,z$ can exist.

The first assumption is not necessary, your proof can be extended to remove this assumption.
This was the basis of Lam\'e's ``proof" of FLT in 1847, which relied on (2); unfortunately (2) holds only for $p\le 19$.
Kummer later generalized Lam\'e's argument to many cases where $\Z[\zeta_p]$ is not a UFD; Kummer's argument applies whenever the order of  ideal class group of the ring of integers of $\Q(\zeta_p)$ is not divisible by $p$, which is expected to hold for infinitely many $p$ (the set of so-called \emph{regular} primes is believed to be infinite but this is not known).

For the sake of concreteness, let us fix an embedding of $\Q(\zeta_p)$ in $\C$ by defining $\zeta_p:=e^{2\pi i/p}$,
and for any $z\in \Q(\zeta_p)\subseteq \C$, let $\bar z$ denote its complex conjugate.
If $S$ is a set, then $a\equiv b \bmod S$ means $a-b\in S$.

\begin{enumerate}
\setlength\itemsep{0pt}

\item[\textbf{(a)}] Show that $\zeta_p^i-\zeta_p^j$ properly divides $p$ in the ring $\Z[\zeta_p]$ for any $i\not\equiv j\bmod p$.
\item[\textbf{(b)}] Show that if a non-unit $\alpha\in \Z[\zeta_p]$ divides $x+y\zeta_p^i$ then it does not divide $x+y\zeta^j_p$ for any $j\not\equiv i\bmod p$.
\item[\textbf{(c)}] Show that $x+y\zeta^i_p=u_i\alpha_i^p$ for some $\alpha_i\in \Z[\zeta_p]$ and $u_i\in \Z[\zeta_p]^\times$.
\item[\textbf{(d)}] Prove that $1+t+\cdots +t^{p-1}$ is irreducible in $\Q[t]$; conclude that $\{1,\zeta_p,\ldots,\zeta_p^{p-2}\}$ is a basis for $\Z[\zeta_p]$ as a $\Z$-module.
\item[\textbf{(e)}] Show that in any commutative ring $A$ we have $\alpha^p+\beta^p\equiv(\alpha+\beta)^p\bmod pA$ for all $\alpha,\beta\in A$.
\item[\textbf{(f)}] Let $\alpha\in\Z[\zeta_p]$. Show (1) $\alpha^p\equiv a\bmod p\Z[\zeta_p]$ for some $a\in\Z$, (2) $\alpha^p\equiv \bar\alpha^p\bmod p\Z[\zeta_p]$, (3) $p\not\in \Z[\zeta_p]^\times$, and (4) if $u\in \Z[\zeta_p]^\times$ then $u/\bar u\ne -\zeta_p^i$ for any $i$.
\item[\textbf{(g)}] Show that if $\alpha\in \Qbar^\times$ is an algebraic integer whose Galois conjugates all lie in the unit disk in $\C$ then $\alpha$ is a root of unity.
\item[\textbf{(h)}] Show that if $u\in \Z[\zeta_p]^\times$ then $u/\bar u = \zeta_p^i$ for some $i$.
\item[\textbf{(i)}] Show that if $x+y\zeta_p\equiv u\alpha^p\bmod p\Z[\zeta_p]$ with $u\in \Z[\zeta_p]^\times$, then for some $0\le j\le p-1$ we must have $x+y\zeta_p\equiv (x+y\zeta_p^{-1})\zeta_p^j\bmod p\Z[\zeta_p]$.
\item[\textbf{(j)}] Show that $x+y\zeta_p\equiv (x + y\zeta_p^{-1})\zeta_p^j\bmod p\Z[\zeta_p]$ only if $j\equiv 1\bmod p$.
\item[\textbf{(k)}] Show that if $x+y\zeta_p\equiv x\zeta_p+y\bmod p\Z[\zeta_p]$ then $x\equiv y\bmod p$.
\item[\textbf{(l)}] Assuming $\Z[\zeta_p]$ is a UFD, show $x^p+y^p=z^p$ has no solutions with $xyz\not\equiv 0\bmod p$.
\end{enumerate}

\subsection*{Problem 3. Factoring primes in quadratic fields (32 points)}
This is a follow-up to Problem 3 on Problem Set 1.  Let $p,q\in \Z$ denote primes.
\begin{enumerate}
\item[\textbf{(a)}] Let $K$ be a quadratic extension of $\Q$ with ring of integers $\O_K$, and let
\[
(q)=\q_1^{e_1}\cdots\q_n^{e_n}
\]
be the unique factorization of the principal ideal $(q)$ in $\O_K$.  Show that
\[
[\O_K:q\O_K]=q^2=\prod_{i=1}^n [\O_K:\q_i]^{e_i},
\]
(where $[B:A]$ denotes the index of $A$ in $B$ as an additive abelian group),
and conclude that there are three possibilities: $(q)$ is prime, $(q)=\q_1\q_2$, or $(q)=\q_1^2$.
\item[\textbf{(b)}] For $K:=\Q(\sqrt{p})$ determine the unique factorization of $(q)$ in $\O_K$ explicitly; that is, determine which of the three possibilities admitted by (a) occurs and when applicable, write $\q_i$ in the form $(q,\alpha_i)$ for some explicitly described $\alpha\in \O_K$.  Be sure to address the cases $q=2$ and $q=p$ which may require special treatment.
\item[\textbf{(c)}] Do the same for $K:=\Q(\sqrt{-p})$.
\item[\textbf{(d)}] For primes $p,q\ne 2$, let $K:= \Q(\sqrt{\pm p})$ and relate the factorization of $(q)$ in $\O_K$ you determined in parts (b) and (c) to the factorization of $x^2\mp p$ in $\Fq[x]$.
\end{enumerate}

\subsection*{Problem 4. Computing the norm and trace (32 points)}
Let $L/K$ be a finite extension of fields, let $\Kbar$ be an algebraic closure of $K$ containing $L$, and define $\Sigma:=\Hom_K(L,\Kbar)$.
\begin{enumerate}
\item[\textbf{(a)}] Prove that for all $\alpha\in L$ we have
\[
N_{L/K}(\alpha)=\left(\prod_{\sigma\in\Sigma}\sigma(\alpha)\right)^{[L:K]_i}\qquad\text{and}\qquad T_{L/K}(\alpha)=[L:K]_i\left(\sum_{\sigma\in\Sigma}\sigma(\alpha)\right).
\]
\end{enumerate}
Fix $\alpha\in L^\times$ with minimal polynomial $f(x)=\sum a_ix^i$ over $K$, and let $f(x)=\prod_{i=1}^d(x-\alpha_i)$ be the factorization of $f$ in $\Kbar[x]$.
Define $n:=[L:K]$ and $e:=[L:K(\alpha)]$ (so $de=n$).

\begin{enumerate}
\item[\textbf{(b)}] Prove that
\[
\N_{L/K}(\alpha)=\prod_{i=1}^d\alpha_i^e=(-1)^na_0^e\qquad\text{and}\qquad\T_{L/K}(\alpha)=\sum_{i=1}^de\alpha_i=-ea_{d-1}.
\]
\item[\textbf{(c)}] Prove that $T_{L/K}=0$ (as a linear map) if and only if $L/K$ is inseparable.
\end{enumerate}
\smallskip

\subsection*{Problem 5. Survey (4 points)}
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
9/11 & Properties of Dedekind domains & & & & \\\hline
9/16 & \'Etale algebras, norm and trace & & & & \\\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}{9}
\bibitem{lorenzini}
Dino Lorenzini, \href{http://bookstore.ams.org/gsm-9/}{\textit{An invitation to arithmetic geometry}}, Graduate Studies in Mathematics \textbf{9}, American Mathematical Society, 1996.
\end{thebibliography}

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