\classheader{2013-01-07}

The main subject of this course is commutative ring theory, and its relation to algebraic number theory and algebraic geometry. We will also see some more advanced topics such as class field theory and the Weil conjectures, though we will not go into them in depth.

There will be no exams. The assignments will appear on each Friday, on the Chalk site if it can be set up; otherwise, on each Thursday evening, the assignments will be sent by email. They will then be due the following Friday.

If you look on the street, you never meet a commutative ring; that's rather strange. They are rather shy I think. We need to ask them to come to this room. Rings, rings, please come! Rings, rings please come! \textit{*shuffles along the floor, playing the part of the ring*} Finite fields, come! Rings of functions, come! \textit{*hops*} I think they are here now.

\subsection*{The Mysterious Analogy Between Prime Numbers and Points}

Let $A$ be a commutative ring. We define
\[\max(A)=\{\text{maximal ideals of }A\}.\]
There is a bijection
\begin{center}
\begin{tikzcd}[row sep=-0.03in]
\max(\Z) \ar[leftrightarrow]{r} & \{\text{prime numbers}\}\\
\upin & \upin\\
(p)=p\Z & p
\end{tikzcd}
\end{center}
There is also a bijection
\begin{center}
\begin{tikzcd}[row sep=-0.03in]
\max(\C[T]) \ar[leftrightarrow]{r} & \C\\
\upin & \upin\\
(T-\alpha) & \alpha
\end{tikzcd}
\end{center}
More generally, there is a bijection
\begin{center}
\begin{tikzcd}[row sep=-0.03in]
\max(\C[T_1,\ldots,T_n]) \ar[leftrightarrow]{r} & \C^n\\
\upin & \upin\\
(T_1-\alpha_1,\ldots,T_n-\alpha_n) & (\alpha_1,\ldots,\alpha_n)
\end{tikzcd}
\end{center}
Note that $(T_1-\alpha_1,\ldots,T_n-\alpha_n)=\{f\in\C[T_1,\ldots,T_n]\mid f(\alpha_1,\ldots,\alpha_n)=0\}$.

We can walk on $\C$, but I think it is hard walking on the prime numbers. I hope someday I can find a pair of shoes that can help. One difficulty is that the points in $\C$ are all the same size, but the primes are like stones of different sizes; 3 is a little bigger than 2, 5 is a little bigger than 3$\ldots$

Let $X$ be a compact Hausdorff space, and let $A=C(X)=\{\text{continuous maps }X\to\R\}$. Then there is a bijection
\begin{center}
\begin{tikzcd}[row sep=-0.03in]
\max(A) \ar[leftrightarrow]{r} & X\\
\upin & \upin\\
\{f\in A\mid f(p)=0\} & p
\end{tikzcd}
\end{center}
If $X$ is not compact, this is false. We can see this by considering $X=\R$ for example. In $A=\{\text{continuous maps }\R\to\R\}$, we can consider the ideal
\[I=\{f\in A\mid \text{there is some }c\text{ such that }f(x)=0\text{ if }x>c\}.\]
There is a proposition from commutative ring theory:
\begin{proposition}
If $A$ is a commutative ring and $I\subsetneq A$ is a proper ideal of $A$, then there is some $\frak{m}\in\max(A)$ such that $I\subset \frak{m}$.
\end{proposition}
This proposition requires the Axiom of Choice, so we will skip the proof for the moment. But the proposition implies that there is some maximal ideal of $A$ containing $I$, and no ideal of the form $\{f\in A\mid f(p)=0\}$ can contain $I$, so there must be other maximal ideals.

Now consider $X=\{(x,y)\in\C^2\mid y^2=x^3+1\}$, and let $A=\{\text{polynomial functions on }X\}$. For example, the function $x$ sends $(a,b)\in X$ to $a\in\C$, and the function $y$ sends $(a,b)\in X$ to $b\in\C$. Note that
\[A=\{\C\text{-valued functions on }X\text{ written as a polynomial over }\C\text{ in the functions }x,y\}.\]
Then $A\cong \C[T,\sqrt{T^3+1}]$, where $T$ is the function $x$. This ring is a quadratic extension of $\C[T]$, which you should think of as being similar to an extension of $\Z$, such as for example $\Z[\sqrt{26}]$.

There is an isomorphism $A\cong\C[T_1,T_2]/(T_2^2-T_1^3-1)$, where $T_1\mapsto T$ and $T_2\mapsto\sqrt{T^3+1}$. There is also a bijection
\begin{center}
\begin{tikzcd}[row sep=-0.03in]
\max(A) \ar[leftrightarrow]{r} & X\\
\upin & \upin\\
\{f\in A\mid f(p)=0\} & p
\end{tikzcd}
\end{center}
which can be deduced from the correspondence between $\max(\C[T_1,T_2])$ and $\C^2$,
\begin{center}
\begin{tikzcd}[row sep=-0.03in]
\max(\C[T_1,T_2]) \ar[leftrightarrow]{r} & \C^2\\
\upsubset & \upsubset\\
\max(A)   \ar[leftrightarrow]{r} & X
\end{tikzcd}
\end{center}
(Recall that there is a bijection
\[\max(A/I)\longleftrightarrow \{\frak{m}\in\max(A)\mid I\subseteq \frak{m}\},\]
where $M\in\max(A/I)$ corresponds to $\{x\in A\mid x\bmod I\in M\}$.)

In the ring $\Z[\sqrt{-26}]$, note that we do not have unique factorization: \[3^3=27=(1+\sqrt{-26})(1-\sqrt{-26}).\]
Writing $\frak{p}=(3,1+\sqrt{-26})$ and $\frak{p}'=(3,1-\sqrt{-26})$, we can recover unique factorization for ideals:
\[(3)=\frak{p}\frak{p}',\quad (1+\sqrt{-26})=\frak{p}^3,\quad (1-\sqrt{-26})={\frak{p}'}^3\]
\[(\frak{p}\frak{p}')^3=(27)=\frak{p}^3{\frak{p}'}^3.\]
(recall that for ideals $I$ and $J$, their product is $IJ=\{\sum_{i=1}^n a_ib_i\mid a_i\in I, b_i\in J\}$.)

There is a similar situation in $A$: we have $x^3=(y+1)(y-1)$. Define \[\frak{p}=(x,y+1)=\{f\in A\mid f(0,-1)=0\}\longleftrightarrow(0,-1)\in X,\]
and similarly $\frak{p}'=(x,y-1)\longleftrightarrow(0,1)\in X$. Then $(x)=\frak{p}\frak{p}'$, $(y+1)=\frak{p}^3$, and $(y-1)={\frak{p}'}^3$.
% http://tex.stackexchange.com/questions/18359
\begin{center}
\begin{tikzpicture}
\begin{axis}[title={$y^2=x^3+1$}, no markers, axis lines=middle,axis y line=center,axis equal,axis y line=middle, axis x line=middle, xmin=-2.5, xmax=2.5, ymin=-2.5, ymax=2.5,xtick={-2,-1,...,2},ytick={-2,-1,...,2},axis equal,width=7cm, height=7cm,tick label style={color=gray}]
\addplot +[no markers,raw gnuplot,thick] gnuplot {
set contour base;
set cntrparam levels discrete 0.00;
unset surface;
set view map;
set isosamples 1000;
set xrange[-4:3];
set yrange[-3:3];
splot y^2-x^3-1;
};
\end{axis}
\end{tikzpicture}
\end{center}

Observe that the function $x$ has a zero of order 1 at $(0,1)$ and $(0,-1)$, that $y+1$ has a zero of order 3 at $(0,-1)$, and $y-1$ has a zero of order 3 at $(0,1)$.

Kummer was the one who realized that even though there is no unique factorization in the world of numbers, it could be recovered in the world of ideals. This observation was then imported to the world of geometry.


As we've seen, there is an analogy between $\Z$ and $\C[T]$. In fact, the analogy between $\Z$ and $\F_p[T]$ is even stronger; for example the theory of zeta functions is very similar for $\Z$ and $\F_p[T]$. We don't know the true reason why they are so similar; perhaps they are children of the same parents. But we don't know who their parents are; their parents are missing.

In 1912, Wegener compared the west coast of Africa and the east coast of South America,

\begin{center}
\rotatebox{350}{\includegraphics[scale=0.5]{southamerica.jpg}}\raisebox{0.5in}{\rotatebox{330}{\includegraphics[scale=0.5]{africa.jpg}}}
\end{center}

and hypothesized that at one point they were connected. It took a long time for his theory to be accepted.