\documentclass[12pt]{article}
\usepackage{amsfonts, amsthm, amsmath}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.5in}
\setlength{\topmargin}{0in}
\setlength{\headheight}{0in}
\setlength{\headsep}{0in}
\setlength{\parskip}{0pt}
\setlength{\parindent}{20pt}

\def\CC{\mathbb{C}}
\def\FF{\mathbb{F}}
\def\PP{\mathbb{P}}
\def\QQ{\mathbb{Q}}
\def\RR{\mathbb{R}}
\def\ZZ{\mathbb{Z}}
\def\gotha{\mathfrak{a}}
\def\gothm{\mathfrak{m}}
\def\gotho{\mathfrak{o}}
\def\gothp{\mathfrak{p}}
\DeclareMathOperator{\disc}{Disc}
\DeclareMathOperator{\Norm}{Norm}
\DeclareMathOperator{\Cl}{Cl}

\def\head#1{\medskip \noindent \textbf{#1}.}
\def\fixme#1{\textbf{FIXME! #1}}

\begin{document}

\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lectures 15-16: Completions and Local Fields \\
October 1 and 3, 2001
\end{center}

\head{Reference}
Neukirch, Sections 2.4 and 2.5, and a bit of 2.6.

\head{Jargon watch}
An exponential valuation is \emph{discrete} if it takes values in a discrete
subgroup of $\RR$, which can always be normalized to be $\ZZ$ itself.
A valuation is \emph{discrete} if it is nonarchimedean and comes from
a discrete exponential valuation. (For example, the $p$-adic valuation on
$\QQ$ is discrete.)

Recall that given a field with a nonarchimedean valuation, the
\emph{valuation ring} of the field is the set of elements of norm at most 1.

\head{Outline of lectures}

\begin{enumerate}
\item
For $K$ a number field and $\gothp$ a nonzero prime ideal,
define $K_{\gothp}$ as the completion of $K$ with respect to the
valuation $|\cdot|_{\gothp}$. If $\gothp \cap \QQ = (p)$, then
$[K_{\gothp}:\QQ_p] \leq [K:\QQ]$, but the inequality can be strict.
\item
Demonstrate some examples for the
Gaussian numbers $\QQ(i)$, with $\gothp = (2+i)$, $\gothp = (3)$,
and the ``anomalous'' case $\gothp = (1+i)$.
\item
Define a \emph{local field} as a field which is complete with respect to a 
discrete valuation, with finite residue field.
Note: the \emph{only}
such fields are finite extensions of $\QQ_p$ or of the power
series field $\FF_p((t))$.
\item
Focus a bit on discrete valuation rings. (Remember, these aren't necessarily
complete.)
Note that the maximal ideal $\gothm$
of a DVR is generated by any element of $\gothm \setminus \gothm^2$, and
every ideal of a DVR is a power of the maximal ideal. In particular, a DVR
is a principal ideal domain. For $U_i$ the group of units congruent to
$1$ modulo $\pi^i$, point out the isomorphisms $U_0/U_1 \cong
(R/\gothm)^*$ and $U_i/U_{i+1} \cong R/\gothm$ for $i \geq 1$.
\item
State and prove 
Hensel's lemma: if a monic
polynomial over the ring of integers in
a local field factors, modulo the maximal ideal, into two coprime
factors, then there is a corresponding factorization of the original 
polynomial.
Example from last time: if $n$ is a quadratic residue modulo $p$, then $x^2-n$ factors
over $\ZZ_p$.
\item
State the extension theorem: if $L/K$ is a finite extension of fields, and
$K$ is complete with respect to the valuation $|\cdot|$, then there exists
a unique extension of $|\cdot|$ to $L$. Sketch the proof in the nonarchimedean
case (the archimedean case uses Ostrowski's theorem, see Neukirch 2.4).
\item
Compare the previous theorem
to the case $\QQ(i)/\QQ$ of an extension of number fields:
a valuation downstairs may extend upstairs in multiple ways.
Sneak preview:
if $K$ is a number field and $\gothp$ is a prime ideal of $\gotho_K$,
the extensions of $|\cdot|_\gothp$ to a finite extension $L$ of $K$ correspond
to the prime factors of $\gothp \gotho_L$.
\item
Define the Newton polygon of a polynomial over a discrete valuation ring.
Show that the slopes of the Newton polygon coincide with the valuations of
the roots of the polynomial.
\item
If $R$ is the valuation ring of a local field, observe that any irreducible
polynomial over $R$ only has a single slope in its Newton polygon
(by the extension theorem).
\end{enumerate}

\end{document}