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\begin{center}
\bf
Math 254A, UC Berkeley, Fall 2001 (Kedlaya) \\
Lecture 1: The Gaussian Integers \\
August 27, 2001
\end{center}

\head{Reference} Neukirch, Section 1.1.

\head{Jargon watch} A ring $R$ is \emph{euclidean} if there exists
a function $f: R \to \ZZ_{\geq 0}$ such that:
\begin{enumerate}
\item[(a)] For $\alpha \in R$, $f(\alpha) = 0$ if and only if $\alpha = 0$.
\item[(b)] For $\alpha, \beta \in R$, $f(\alpha \beta) = f(\alpha) f(\beta)$.
%\item[(b)] For any $c>0$, there are only finitely many elements $\alpha$ of $R$
%such that $f(\alpha) < c$.
\item[(c)] For any $\alpha, \beta \in R$, with $\beta$ nonzero, there exist
$\gamma, \delta \in R$ with $\alpha = \beta \gamma + \delta$ and
$f(\delta) < f(\beta)$.
\end{enumerate}
For example, take $R = \ZZ$ and $f(x) = |x|$.

A \emph{unit} of a ring $R$ is an element with a multiplicative inverse.
An element $\alpha$ of a ring $R$ is \emph{irreducible} if 
it not a unit, and whenever
$\beta$ and $\gamma$ are elements of $R$ whose product is $\alpha$,
one of $\beta$ or $\gamma$ is a unit. (Neukirch also calls these elements
``prime'', but this is a bad habit, as we'll see later.)
A ring $R$ is \emph{factorial} if it has no zero divisors (i.e., $\alpha
\beta = 0$ implies $\alpha = 0$ or $\beta = 0$),
every element can be factorized as a product of irreducible elements, and this 
factorization is unique up to multiplying elements by units.
For example, $\ZZ$ is factorial, as is the ring $\CC[x]$ of polynomials
in one variable over the complex numbers (or any other field, for that matter).

\head{Outline of lecture}
\begin{enumerate}
\item Do administrative stuff: describe course, homework, exams, grading.
Hand out most recent version of course description, and Problem~Set~1.
Then get to work.
\item Recall statement of Fermat's two squares theorem in the prime case.
Point out that $p = a^2+b^2$ iff $p = (a+bi)(a-bi)$.
\item Define ring $\ZZ[i]$. Note that the only units are $\pm 1$, $\pm i$.
\item Prove $\ZZ[i]$ is euclidean.
\item
Note that for $R$ euclidean, for any $x,y \in R$, the set
$\{\alpha x + \beta y: \alpha, \beta \in R\}$ has an element that divides
all of the others.
\item Conclude that $p \equiv 1 \pmod{4}$ is the sum of two squares. Yay!
\item
Point out that $R$ euclidean implies that every ideal in $R$ is principal,
and that every ideal being principal implies unique factorization.
\end{enumerate}

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