Applications of class number

There is the notion of class number from algebraic number theory. Why is such a notion defined and what good comes out of it?

It is nice if it is $1$; we have unique factorization of all ideals; but otherwise?

Solution 1:

As others have said, often what you want for a particular Diophantine application is that the class number of a certain number field be relatively prime to a certain number. The famous example of this (as already noted by others) is Kummer's Theorem that for an odd prime $p$, the Fermat equation $x^p + y^p = z^p$ has no integer solutions with $xyz \neq 0$ if the ring of integers of $\mathbb{Q}[e^{\frac{2 \pi i}{p}}]$ has class number prime to $p$.

Another -- simpler -- nice example is the Mordell equation $y^2 + k = x^3$. If $k \equiv 1,2 \pmod 4$ and the ring $\mathbb{Z}[\sqrt{-k}]$ has class number prime to $3$, then all of the integer solutions to the Mordell equation can be found. See Section 4 of

http://math.uga.edu/~pete/4400MordellEquation.pdf

for an exposition of this which is (I hope) reasonably elementary and accessible to undergraduates.

Solution 2:

The class group of a number field $K$ can be used to parametrize other objects.

1) If $[L:K] = n$, the possible $O_K$-module structure of $O_L$ is described by the ideal classes of $K$, although it is still an open question in general to show for each $n > 1$ and each ideal class of $K$ that there's an extension $L/K$ with degree $n$ such that $O_L$ as an $O_K$-module corr. to that ideal class. (This is known for small $n$, but not for general $n$.)

2) The orbits of the action of $\text{SL}_2(O_K)$ on ${\mathbf P}^1(K)$ are in bijection with ideals classes in $K$. For instance, the action is transitive iff $K$ has class number 1.

3) When $O$ is a quadratic order with discriminant $d$, the (narrow) class group of $O$ describes the primitive quadratic forms of discriminant $d$ up to proper equivalence. Here we need a slightly more general concept than the usual ideal class group (unless $O = O_K$).

4) Weierstrass equations for an elliptic curve over $K$ up to a standard change of variables are related to ideal classes in $K$ (see Silverman's first book on ell. curves, Chap. VIII).