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\begin{document}
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\large
\textbf{18.785 Number Theory\hspace{228pt}Fall~2018}\\\vspace{4pt}
\textbf{Problem Set \#3\hspace{228pt}Due: 10/1/2018}
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\subsection*{Description}

These problems are related to the material in Lectures 5--7.
Your solutions should be written up in late and submitted as a pdf-file named \texttt{SurnamePset3.pdf} (replace \texttt{Surname} with your surname) via e-mail to \texttt{drew@math.mit.edu} by \textbf{noon} on the date due.
Collaboration is permitted/encouraged, but you must identify your collaborators, and any references consulted other than the lecture notes.
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 the problem set or lecture notes will receive 1--5 points of extra credit.
\medskip

\noindent
\textbf{Instructions:} First do the warm up problems, then pick any combination of problems 1--6 that sums to 96 points and write up your answers in latex.  Finally, be sure to complete the survey problem 7.

\subsection*{Problem 0. Warmup (0 points)}
These warmup exercises do not need to be written up or turned in.
\begin{enumerate}
\setlength\itemsep{0pt}
\item[\textbf{(a)}] Show that odd primes $p$ split over $\Q(\sqrt{d})$ if and only if $x^2-d$ splits in $\Fp[x]$, but that this holds for $p=2$ only when $d\not\equiv 1\bmod 4$.  Then show that for $d\equiv 1\bmod 4$ using $x^2-x+(1-d)/4$ instead of $x^2-d$ works for every prime $p$.
\item[\textbf{(b)}] Let $\O_K$ be the ring of integers of an imaginary quadratic field $K$ and let $c$ be a positive integer.
Prove that $\O:=\Z+c\O_K$ is an order with conductor $c\O_K$ and that $c=[\O_K:\O]$ (the index of $\O$ in $\O_K$ as additive abelian groups).
\item[\textbf{(c)}] Let $L/K$ be a finite Galois extension of number fields.  Prove that if $K$ has any inert primes then $\Gal(L/K)$ is cyclic (as we shall prove later, the converse holds).
\item[\textbf{(d)}] Let $L/K$ be a finite extension of number fields.  Show that a prime of $K$ splits completely in $L$ if and only if it splits completely in the normal closure of $L/K$.
\end{enumerate}

\subsection*{Problem 1. Factoring primes in cubic fields (32 points)}
Let $K=\Q(\sqrt[3]{5})$.
\begin{enumerate}
\item[\textbf{(a)}] Prove that $\O_K=\Z[\sqrt[3]{5}]$.
\item[\textbf{(b)}] Factor the primes $p=2,3,5,7,11,13$ in $\Q(\sqrt[3]{5})$.  Write the prime ideals $\q$ appearing in your factorizations in the form $(p,f(\sqrt[3]{5}))$ where $f\in \Z[x]$ has degree at most 2.
\item[\textbf{(c)}] Prove that the factorization patterns you found in (b) represent every possible case; that is, every possible sum $[K:\Q]=\sum_{\q|(p)} e_\q f_\q$ that can arise for this particular field $K$.  You should find that there is one numerically possible case that does not occur for $p\le 13$; you need to prove that it cannot occur for any prime $p$.
\item[\textbf{(d)}] Find a different cubic field of the form $K=\Q(\sqrt[3]{n})$ for which the one factorization pattern missing from (c) does occur (demonstrate this explicitly).
\end{enumerate}


\subsection*{Problem 2. Factoring primes in cyclotomic fields (32 points)}
Let $\ell$ be a prime and let $\zeta_\ell$ denote a primitive $\ell$th root of unity.
\begin{enumerate}
\item[\textbf{(a)}] Prove that $\Q(\zeta_\ell)/\Q$ is a Galois extension.
\item[\textbf{(b)}] Prove that $\Z[\zeta_\ell]$ is the ring of integers of $\Q(\zeta_\ell)$.
\item[\textbf{(c)}] For each prime $p\ne \ell$, determine the number $g_p$ of primes $\q$ of $\Q(\zeta_\ell)$ lying above~$(p)$, the ramification index $e_p$ and the residue field degree $f_p$ (as a function of $p$ and $\ell$).
\item[\textbf{(d)}] Do the same for $p=\ell$.
\end{enumerate}

\subsection*{Problem 3. Non-monogenic fields (32 points)}
Recall that a number field $K$ is said to be monogenic if its ring of integers $\O_K$ is of the form $\Z[\alpha]$ for some $\alpha\in \O_K$.
Every number field of degree 2 is monogenic; indeed, for $K=\Q(\sqrt{-d})$ we can take $\alpha=(d\pm \sqrt{-d})/2$.
In this problem you will prove that infinitely many number fields of degrees 3 and 4 are not monogenic.
\begin{enumerate}
\item[\textbf{(a)}] Let $K$ be a number field of degree $n>2$ in which the prime $2$ splits completely (so $2\O_K$ is the product of $n$ distinct prime ideals).  Prove that $K$ is not monogenic.
\item[\textbf{(b)}] Prove that if $2$ splits completely in number fields $K_1$ and $K_2$ then it also splits completely in their compositum (the smallest number field containing $K_1$ and $K_2$).
\item[\textbf{(c)}] Show that if $p\equiv \pm 1\bmod 8$ is prime, then $2$ splits completely in $\Q(\sqrt{\pm p})$ (with the same sign in both $\pm$).
Conclude that for each $k\ge 2$, infinitely many number fields of degree $2^k$ are not monogenic and give a quartic example.\footnote{You may assume Dirichlet's theorem on primes in arithmetic progressions, which we will prove later in the course: for any coprime $a,m\in\Z$ there are infinitely many primes $p\equiv a\bmod m$.}
\item[\textbf{(d)}] Consider $K=\Q(\sqrt[3]{ab^2})$, with $a,b\in \Z$ coprime, squarefree, and $a^2\not\equiv b^2\bmod 9$.  Dedekind showed that $(1,\sqrt[3]{ab^2},\sqrt[3]{a^2b})$ is a $\Z$-basis for $\O_K$.  Show that for every $\alpha\in \O_K-\Z$, the index $[\O_K:\Z[\alpha]]$ has the form $ar^3-bs^3$, with $r,s\in\Z$.  Show that infinitely many cubic number fields are not monogenic and give an example.
\end{enumerate}

\subsection*{Problem 4. Orders in Dedekind domains (32 points)}
Let $\O$ be an order (noetherian domain of dimension one with nonzero conductor) with integral closure $B$ (a Dedekind domain) and conductor $\c$ (largest $B$-ideal in $\O$).
\begin{enumerate}
\item[\textbf{(a)}] Prove that for a prime $\p$ of $\O$ the following are equivalent:
\begin{enumerate}[(1)]
\setlength{\itemsep}{2pt}
\item $\p$ does not contain $\c$;
\item $\O=\{x\in B:x\p\subseteq \p\}$;
\item $\p$ is invertible (as a fractional $\O$-ideal);
\item $\O_\p$ is a DVR;
\item $\p\O_\p$ is a principal $\O_\p$-ideal.
\end{enumerate}
Then show that these equivalent conditions all imply that $\p B$ is a prime $B$-ideal.
\item[\textbf{(b)}] Prove that nonzero fractional ideals $I$ of $\O$ prime to $\c$ are invertible, but the converse need not hold (give an explicit counterexample).
\item[\textbf{(c)}] Let $K\ne \Q$ be a number field with ring of integers $\O_K$, let $c\in \Z_{>1}$, and let
\[
\O:=\Z+c\O_K=\{a+b:a\in \Z, b\in c\O_K\}.
\]
Prove that $\O$ is an order with integral closure $\O_K$ and conductor $c\O_K$, and that $c\O_K$ is not principal as an $\O$-ideal.
\item[\textbf{(d)}] Let $K:=\Q(i)$ with $\O_K=\Z[i]$, let $p$ be any prime, and let $\O:=\Z+pi\Z$.
Show that the conductor of $\O$ is $\p:=p\Z+pi\Z$, that $\p$ is a prime $\O$-ideal, and that $\a:=p^2\Z+pi\Z$ is an $\O$-ideal contained in $\p$ but not divisible by $\p$.
\end{enumerate}

\subsection*{Problem 5. A relative extension without an integral basis (32 points)}
Let $K$ be the quadratic field $\Q(\sqrt{-6})$ with ring of integers $A=\Z[\sqrt{-6}]$, let $L:=K(\sqrt{-3})$ be a quadratic extension, and let $B$ be the integral closure of $A$ in $L$ (so $AKLB$ holds).
\begin{enumerate}
\item[\textbf{(a)}] Let $\zeta_3:=\frac{-1+\sqrt{-3}}{2}$. Show that $\{1, \sqrt{2}, \zeta_3\}$ generates $B$ as an $A$-module.
Conclude that $B$ is a torsion free $A$-module, and that if it is a free $A$-module, it has rank~2.
\item[\textbf{(b)}] Show that if $B\simeq A^2$, then $\{1,\zeta_3\}$ is an $A$-module basis for~$B$ (hint: show that if $\{\beta_1,\beta_2\}$ is any $A$-module basis for $B$, then the matrix that expresses $\{1,\zeta_3\}$ in terms of this basis is invertible; to do so you may also want to write $\{1,\sigma(\zeta_3)\}$ in terms of $\{\sigma(\beta_1),\sigma(\beta_2)\}$ with $\sigma\in\Gal(L/K)$).
\item[\textbf{(d)}] Show that $\{1,\zeta_3\}$ is \emph{not} an $A$-module basis for $B$ by showing that you cannot write $\sqrt{2}$ in terms of this basis.  Conclude that $B$ is not a free $A$-module and that the ideal class group $\cl(A):=\I_A/\mathcal{P}_A$ is non-trivial.
\item[\textbf{(d)}] Show that the $A$-module $B$ is isomorphic to the $A$-module $I_1 \oplus I_2$, where $I_1,I_2\in \mathcal{I}_A$ are the fractional $A$-ideals $I_1:=(\zeta_3)$ and $I_2:=\frac{1}{\sqrt{-3}}(3,\sqrt{-6})$.
\end{enumerate}

\subsection*{Problem 6. Modules over Dedekind domains (64 points)}
Let us recall some terminology from commutative algebra.
Let $A$ be a ring and let $M$ be an $A$-module.
A \emph{splitting} of a surjective $A$-module homomorphism $\psi\colon N\to M$ is an $A$-module homomorphism $\phi\colon M\to N$ such that $\psi\circ\phi$ is the identity map; we then have
\[
N=\phi(M)\oplus\ker(\psi)\simeq M\oplus\ker(\psi).
\]
We say that $M$ is \emph{projective} if every surjective $A$-module homomorphism $\psi\colon N\to M$ admits a splitting $\phi\colon M\to N$.
A \emph{torsion} element $m\in M$ satisfies $am=0$ for some nonzero $a\in A$.
If $M$ consists entirely of torsion elements then it is a \emph{torsion module}.
If $M$ has no nonzero torsion elements then it is \emph{torsion free}.  Note that the zero module is a torsion-free torsion module.

Now let $A$ be a Dedekind domain with fraction field $K$.
\begin{enumerate}
\item[\textbf{(a)}] Prove that every finitely generated torsion $A$-module $M$ is isomorphic to
\[
A/I_1\oplus\cdots\oplus A/I_n,
\]
for some nonzero $A$-ideals $I_1,\ldots, I_n$ (you may use the structure theorem for modules over PIDs).
\item[\textbf{(b)}] Prove that every fractional ideal of $A$ is a projective $A$-module.
\item[\textbf{(c)}] Prove that every finitely generated torsion-free $A$-module $M$ is isomorphic to a finite direct sum of nonzero fractional ideals of $A$ (elements of $\mathcal{I}_A$).
\item[\textbf{(d)}] Prove that every finitely generated $A$-module is isomorphic to the direct sum of a finitely generated torsion module and a finitely generated torsion-free module.
\item[\textbf{(e)}] Show that if $M$ is a finitely generated $A$-module then $M\otimes_A K\simeq K^r$ for some $r\in \Z_{\ge 0}$, and that for $M\in \I_A$ we must have $r=1$.
\item[\textbf{(f)}] Let $M$ be a finitely generated torsion-free $A$-module, and let us fix an isomorphism $\iota\colon M\otimes_A K\overset{\sim}{\longrightarrow} K^n$ that embeds $M$ in $K^n$ via $m\mapsto \iota(m\otimes 1)$.
Let $N$ be the $A$-submodule of $K$ generated by the determinants of all $n\times n$ matrices whose columns lie in $M$.
Prove that $N\in \I_A$ and that its ideal class (its image in the ideal class group $\cl(A):=\I_A/\mathcal{P}_A$) is independent of $\iota$; this is the \emph{Steinitz class} of $M$.
\item[\textbf{(g)}] Prove that for any $I_1,\ldots,I_n\in \I_A$ the Steinitz class of $I_1\oplus\cdots\oplus I_n$ is the ideal class of the product $I_1\cdots I_n$.
\item[\textbf{(h)}] Prove that two finite direct sums $I_1\oplus\cdots\oplus I_m$ and $J_1\oplus\cdots \oplus J_n$ of elements of $\I_A$ are isomorphic as $A$-modules if and only if $m=n$ and the ideal classes of $I_1\cdots I_m$ and $J_1\cdots J_n$ are equal.
\item[\textbf{(i)}] Prove that infinite direct sums $\bigoplus_{i=1}^\infty I_i$ and $\bigoplus_{j=1}^\infty J_j$ of elements of $\I_A$ are always isomorphic as $A$-modules.
\end{enumerate}

\subsection*{Problem 7. 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
Problem 5 & & & \\\hline
Problem 6 & & & \\\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").

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\begin{tabular}{l|l|r|r|r|r|r}
Date & Lecture Topic & Material & Presentation & Pace & Novelty\\\hline
9/25 & Ideal norms, Dedekind-Kummer & & & & \\\hline 
9/27 & Primes in Galois extensions & & & & \\\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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