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\textbf{18.785 Number Theory\hspace{228pt}Fall~2015}\\\vspace{4pt}
\textbf{Problem Set \#3\hspace{220pt}Due: 10/05/2015}
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\subsection*{Description}

These problems are related to the material covered in Lectures 5-7.
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{SurnamePset3.pdf} via e-mail to \texttt{drew@math.mit.edu} by 5pm on the due date.
Collaboration is permitted, 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 any combination of problems 1--5 that sums to 99 points and write up your answers in latex.  Finally, be sure to complete the survey problem 6.

\subsection*{Problem 0.}
These are warm up problems that do not need to be written up or turned in.
\begin{enumerate}
\setlength\itemsep{0pt}
\item[\textbf{(a)}] Prove Lemma 5.3, which states that the ramification index and residue degree are multiplicative in towers.
\item[\textbf{(b)}] 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{(c)}] Let $L/K$ be a finite Galois extension of number fields.  Prove that if there exists an inert prime $\p$ of $K$ then $\Gal(L/K)$ must be cyclic (the converse also holds --- it follows from the Chebotarev density theorem, which we will see later in the course).
\item[\textbf{(d)}] Let $L/K$ be a finite extension of number fields.  Show that a prime $\p$ of $K$ splits completely in $L$ if and only if it splits completely in $M$, where $M/K$ is the normal closure of $L/K$.
\end{enumerate}

\subsection*{Problem 1. Factoring primes in cubic fields (33 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) actually does occur (demonstrate this explicitly).
\end{enumerate}


\subsection*{Problem 2. Factoring primes in cyclotomic fields (33 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 $f_p$ (your answer will depend on a relationship between $p$ and $\ell$ that you need to determine).
\item[\textbf{(d)}] Do the same for $p=\ell$.
\end{enumerate}

\subsection*{Problem 3. Non-monogenic fields (33 points)}
Let $A$ be a Dedekind domain with fraction field~$K$, let $L$ be a finite Galois extension of~$K$, and let $L_1,L_2\subseteq L$ be subfields of $L$ that are Galois over $K$ and generate $L$.
\begin{enumerate}
\item[\textbf{(a)}] Let $\p$ be a prime of $A$ and let $D_\q$ be the decomposition group of a prime $\q$ above~$\p$.  Prove that $\p$ splits completely in $L/K$ if and only if $D_\q$ is the trivial group.
\item[\textbf{(b)}] Prove that if $\p$ splits completely in $L_1$ and $L_2$ then $\p$ splits completely in $L$.
\item[\textbf{(c)}] Now assume $A=\Z$, $K=\Q$ and that $L_1$ and $L_2$ are distinct quadratic extensions of $\Q$ in which $2$ splits completely.  Prove that that $\O_L\ne \Z[\alpha]$ for any $\alpha\in \O_L$.
\item[\textbf{(d)}] Assuming Dirichlet's theorem on primes in arithmetic progressions, show that there are infinitely many nonmonogenic bi-quadratic fields $L$ and give three examples.
\end{enumerate}

\subsection*{Problem 4. A relative extension without an integral basis (33 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)}] Show that, as an $A$-module, $B$ is generated by $\{1, \sqrt{2}, \zeta_3\}$, where $\zeta_3=\frac{-1+\sqrt{-3}}{2}$;
conclude that $B$ is a torsion-free finitely generated $A$-module.
\item[\textbf{(b)}] Show that \emph{if} $B$ is a free $A$-module, then it is a free $A$-module of rank 2.
\item[\textbf{(c)}] Show that \emph{if} $B$ is a free $A$-module of rank 2, then $\{1,\zeta_3\}$ is an $A$-module basis for~$B$ (hint: show 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).
\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.
\item[\textbf{(e)}] Explain why (d) implies that the class group $\cl(A):=\I_A/\mathcal{P}_A$ must be non-trivial.
\item[\textbf{(f)}] 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 5. Modules over Dedekind domains (66 points)}

Let us recall some terminology from commutative algebra.
Let $A$ be a ring.
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 $\psi\circ\phi=1_M$; we then have
\[
N=\phi(M)\oplus\ker(\psi)\simeq M\oplus\ker(\psi).
\]
An $A$-module $M$ is \emph{projective} if every $\psi\colon N\to M$ admits a splitting $\phi\colon M\to N$.

Now let $A$ be a domain with fraction field $K$ and let $M$ be an $A$-module.
An element $m\in M$ is \emph{torsion} if there is a nonzero $a\in A$ for which $am=0$.
The \emph{torsion submodule} $M_\tor$ of $M$ is the set of all torsion elements of $M$.
We say that $M$ is \emph{torsion} if $M_\tor=M$, and $M$ is \emph{torsion free} if $M_\tor=0$ is the zero module, equivalently, the homomorphism $M\to M\otimes_A K$ sending $m$ to $m\otimes 1$ is injective.
Note that the zero module is both torsion and torsion free.

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 ideals $I_1,\ldots, I_n$ of $A$ (you may use the structure theorem for modules over PIDs).
\item[\textbf{(b)}] Prove that every fractional ideal $I$ 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 torsion free $A$-module then $M\otimes_A K\simeq K^n$ for some $n\in \Z_{\ge 0}$.
\item[\textbf{(f)}] Let $M$ be a finitely generated torsion-free $A$-module, and let us fix an isomorphism $\iota\colon M\otimes_A K\overset{\sim}{\rightarrow} K^n$ 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 (image in $\cl(A):=\I_A/\mathcal{P}_A$) 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 (hint: show that both are free $A$-modules of rank $\aleph_0$).
\end{enumerate}

\subsection*{Problem 6. 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
\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
9/29 & Dedekind-Kummer, ideal norms & & & & \\\hline 
10/1 & Splitting 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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