\classheader{2013-01-23}

\section*{Ideal class groups}

Let $A$ be a Dedekind domain, and let $K$ be its field of fractions. Then the non-zero fractional ideals (which we defined earlier) form a group under multiplication. There's no standard notation for this group, but we'll denote it by $I_A$.

The non-zero principal fractional ideals, i.e the ones of the form $(a)=aA$ for some $a\in K^\times$, form a subgroup of $I(A)$, which we will denote by $P(A)$.

\begin{definition}
The class group $\classgroup(A)$ of $A$ is defined to be $I(A)/P(A)$.
\end{definition}

If $A=\ringofintegersof{K}$ where $K$ is a number field, then the class number of $K$ is defined to be $\#\classgroup(A)$. The class number is finite for any number field $K$.


Note that there is a homomorphism $\theta:K^\times\to I(A)$ defined by taking $a$ to $(a)$, and that
\[\ker(\theta)=A^\times,\quad \coker(\theta)=\classgroup(A).\]
These two groups are the most important in number theory. When we understand these, then we are doing well.

\begin{remark}
We can identify
\[\classgroup(A)\longleftrightarrow\left\{\begin{array}{c}
\text{isomorphism classes of non-zero}\\
\text{ideals of }A\text{ as }A\text{-modules}
\end{array}\right\}\]
\end{remark}

The class group is a bitter group and a sweet group. It is bitter because when it is non-trivial it makes a mess. It is sweet because it makes things interesting.

\begin{center}
\begin{tikzpicture}
\node at (0,1.5) {a bitter group};
\draw[thick] (0,0) circle (1);
\draw[thick] (0.25,0.4) arc (180:360:0.1);
\draw[thick] (-0.45,0.4) arc (180:360:0.1);
\draw[thick] (0.25,0.25) arc (180:360:0.1);
\draw[thick] (-0.45,0.25) arc (180:360:0.1);
\draw[thick] (0.4,-0.6) arc (0:180:0.4 and 0.2);
\draw[thick,dotted] (0.35,0.05) to (0.35,-0.3);
\draw[thick,dotted] (-0.35,0.05) to (-0.35,-0.3);
\end{tikzpicture}\qquad \begin{tikzpicture}
\node at (0,1.5) {a sweet group};
\draw[thick] (0,0) circle (1);
\draw[thick] (0.45,0.3) arc (0:180:0.1);
\draw[thick] (-0.25,0.3) arc (0:180:0.1);
\draw[thick] (0.45,0.15) arc (0:180:0.1);
\draw[thick] (-0.25,0.15) arc (0:180:0.1);
\draw[thick] (-0.4,-0.4) arc (180:360:0.4 and 0.2);
\end{tikzpicture}
\end{center}


There is a cake shop in Balmont, which is north of Chicago. The class group is the same as this cake shop; it is a very nice cake shop.


Class groups have mysterious relations with values of zeta functions. In the 19th century, the class number formula was discovered, which connects zeta functions with both the class number and the unit group. Thus, zeta functions are related to two of the most important groups in number theory.

Another result of the 19th century:

\begin{theorem}[Kummer's criterion]
Let $p$ be a prime number. The class number of $\Q(\zeta_p)$ is divisible by $p$ if and only if, for some even integer $2\leq r\leq p-3$, the numerator of $\frac{\zeta(r)}{\pi^r}\in\Q$ is divisible by $p$.
\end{theorem}

Recall that $\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$. Then Euler proved in 1735 that
\[\zeta(2)=\frac{\pi^2}{6},\quad \zeta(4)=\frac{\pi^4}{90},\quad \zeta(6)=\frac{\pi^6}{945}\]
and in general, he proved that for any $n\geq 1$,
\[\zeta(2n)=(-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2\cdot (2n)!}\in\Q\pi^{2n},\]
where the $B_{2n}$ is the $2n$th Bernoulli number. The Bernoulli numbers are defined by
\[\frac{t}{e^t-1}=\sum_{m=0}^\infty\frac{B_m}{m!}t^m.\]
It turns out that the numerator of $\frac{\zeta(r)}{\pi^r}$ is 1 for $r=2,4,6,8,10$. However,
\[\frac{\zeta(12)}{\pi^{12}}=\frac{691}{3^4\cdot 5^3\cdot 7^2\cdot 11\cdot 13}.\]
The primes $p$ such that the class number of $\Q(\zeta_p)$ is divisible by $p$ are
\[37,\;59,\;67,\;101,\;103,\;131,\;\ldots,\; 691,\;\ldots\]

In the 20th century, people discovered deeper relations between zeta functions and arithmetic groups like the ideal class group (there were also many more zeta functions to think about). For example, Iwasawa theory was developed, and there is the conjecture of Beilinson.

Recall that Kummer's two motivations were
\begin{itemize}
\item Hope to prove Fermat's last theorem
\item Hope to have progress on ``generalized reciprocity law'' (we now call this class field theory)
\end{itemize}

\section*{A great stream in number theory}

In the 1630's, Fermat proved for an odd prime $p$, that $p=x^2+y^2$ for some $x$ and $y$ if and only if $p\equiv 1\bmod 4$. For example,
\[5=2^2+1^2,\qquad 13=3^2+2^2,\qquad 17=4^2+1^2,\qquad 89=8^2+5^2\]
In 1796, Gauss proved the quadratic reciprocity law.

In the 19th century, Kummer and others proved the generalized reciprocity law.

Takagi, my advisor's advisor, and Artin in the 1920's worked on class field theory.

From 1965 forward, Langlands worked on Langland's conjectures (non-commutative version of class field theory).

In 1994, Wiles made big progress on the Langland's conjectures, and proved Fermat's last theorem.

Two motivations of Wiles were
\begin{itemize}
\item Hope to prove Fermat's last theorem
\item Hope to have progress on Langlands conjectures
\end{itemize}

Fermat's last theorem has clearly given a lot of energy to mathematicians. In contrast, if Fermat had talked about
\[x^{10}+xy+6345=z^{y!}+yz\]
nobody would have cared.


Recall that $p=x^2+y^2$ implies that $p=(x+yi)(x-yi)$ in $\Z[i]$. 

Kummer generlized this by proving  that $p$ decomposes ``completely'' in $\Z[\zeta_n]$ (into $\phi(n)$ distinct primes) if and only if $p\equiv 1\bmod n$.