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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lecture 14: The $p$-adic Numbers \\
September 28, 2001
\end{center}

\head{Reference}
Neukirch, Sections 2.1 and 2.2.

\head{Outline of lecture}

\begin{enumerate}
\item
Define the $p$-adic integers $\ZZ_p$
naively, as ``infinite series in $p$.''
Demonstrate addition, multiplication, additive inverses, multiplicative
inverses. Define the $p$-adic numbers $\QQ_p$ as infinite series in $p$ with
finitely many places after the decimal point.
\item
Define the $p$-adic integers less naively, as coherent sequences in
$\ZZ/p\ZZ \times \ZZ/p^2\ZZ \times \cdots$.
\item
Define the $p$-adic integers (resp., the $p$-adic numbers)
even less naively, as the completion of
$\ZZ$ (resp., of $\QQ$) under the $p$-adic valuation. (We'll repeat this
more generally in the next lecture.)
\item
Notice that $\ZZ_p$ is a local ring with maximal ideal $p\ZZ_p$. (That is,
every element of $\ZZ_p \setminus p\ZZ_p$ is a unit.) In particular,
any rational $r/s$, where $s$ is coprime to $p$, is a $p$-adic integer
(in fact, a terminating or rational one, and conversely).
\item
Demonstrate some simple cases of Hensel's lemma (simplest form: any monic
polynomial over $\ZZ_p$
with a simple root modulo $p$ has a root in $\ZZ_p$). For example, if $n$ is a 
quadratic residue modulo $p$, then $n$ has two square roots in $\ZZ_p$.
Consequence: the $p$-adic numbers include lots of algebraic numbers!
\item
Note that since $\ZZ_p$ is complete with respect to the $p$-adic valuations,
we can define some functions in terms of convergent power series. For
example, if $p \neq 2$, then $\exp(x) = \sum_{n=0}^\infty x^n/n!$ 
and $\log(1+x) = \sum_{n=1}^\infty (-1)^{n-1} x^n/n$
converge for $x \in p\ZZ_p$
\item
Mention that a polynomial $F(x_1, \dots, x_n)$ has a zero in the $p$-adic
integers if and only if it has a zero modulo every power of $p$.
\end{enumerate}

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