\documentclass[12pt]{article}
\usepackage{fullpage, amsmath, amssymb, stmaryrd, amsthm, hyperref}
\usepackage[all]{xy}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}

\newcommand{\calF}{\mathcal{F}}
\newcommand{\calG}{\mathcal{G}}
\newcommand{\calH}{\mathcal{H}}
\newcommand{\calI}{\mathcal{I}}
\newcommand{\calJ}{\mathcal{J}}
\newcommand{\calO}{\mathcal{O}}
\newcommand{\calL}{\mathcal{L}}
\newcommand{\calU}{\mathcal{U}}
\newcommand{\frakm}{\mathfrak{m}}
\newcommand{\frakp}{\mathfrak{p}}
\newcommand{\frakq}{\mathfrak{q}}

\newcommand{\CC}{\mathbb{C}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\Fp}{\mathbb{F}_p}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\ZZ}{\mathbb{Z}}

\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\Div}{Div}
\DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Norm}{Norm}
\DeclareMathOperator{\red}{red}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\Trace}{Trace}
\DeclareMathOperator{\im}{im}

\begin{document}

\begin{center}
\textbf{Math 204A (Number Theory), UCSD, fall 2020 \\
Problem Set 6 -- due Thursday, November 19, 2020}
\end{center}

\begin{enumerate}

\item
\begin{enumerate}
\item[(a)]
Let $N \geq 3$ be an integer and let $\Phi_N$ be the $N$-th cyclotomic polynomial. Prove that for every integer $m$,
every prime divisor of $\Phi_N(m)$ not dividing $N$ is congruent to 1 modulo $N$. 
\item[(b)]
In class, I stated that for every number field $K$, there exist infinitely many primes $p$ which split completely in $K$.
Prove this for the cyclotomic field $K = \QQ(\zeta_N)$ \emph{without} using Dirichlet or Chebotarev. (Hint: suppose there are only finitely
many, take their product...)
\end{enumerate}

\item
Let $L/K$ be an extension of number fields with Galois closure $M$.  Let $\frakp$ be a prime of $K$.
\begin{enumerate}
\item[(a)]
Prove that if $\frakp$ is unramified in $L$, then it is unramified in $M$ also.
\item[(b)]
Prove that if $\frakp$ is unramified and totally split in $L$, then it is totally split in $M$ also.
\end{enumerate}
(Hint for both parts: put $G = \Gal(M/K)$ and $H = \Gal(M/L)$. Pick a prime $\frakq$ of $M$ above $\frakp$
and consider how $G_{\frakq}$ or $I_{\frakq}$ intersects the various conjugates of $H$.)

\item
Let $m$ be an integer which is not a perfect square, and put $K = \QQ(m^{1/4})$. Let $L$ be the Galois closure of $K/\QQ$.
Let $p$ be an odd prime not dividing $m$ and let $\frakq$ be a prime above $p$ in $L$. For each possible value for the decomposition
group $G_{\frakq}$, describe the corresponding splitting of $p$ in $K$.

\item
\begin{enumerate}
\item[(a)]
Let $k$ be a field. Prove that the ring of formal power series $k \llbracket t \rrbracket$ is a discrete valuation ring.
\item[(b)]
Prove that the subring of $\RR \llbracket t \rrbracket$ consisting of power series with positive radius of convergence
is a discrete valuation ring.
\end{enumerate}

\item
Prove that for every finite abelian group $A$, there exists a Galois number field $K$ with $\Gal(K/\QQ) \cong A$.
(Hint: find $K$ inside a suitable cyclotomic number field. Remember that $A$ is a product of cyclic groups.)

\end{enumerate}

\end{document}
