The statement that the class number measures the failure of the ring of integers to be a ufd is very common in books. ufd iff class number is 1. This inspires the following question:


Is there a quantitative statement relating the class number of a number field to the failure of unique factorization in the maximal order - other than $h = 1$ iff $R$ is a ufd?


In what sense does a maximal order of class number 3 "fail more" to be a ufd than a maximal order of class number 2?


Is it true that an integer in a field of greater class number will have more distinct representations as the product of irreducible elements than an integer in a field with smaller class number?



Class number measuring the failure of unique factorization
Theorem (Carlitz, 1960): The ring of integers $\mathbb{Z}\_F$ of an algebraic number field $F$ has class number at most $2$ iff for all nonzero nonunits $x \in \mathbb{Z}\_F$, any two factorizations of $x$ into irreducibles have the same number of factors.


A proof of this (and a 1990 generalization of Valenza) can be found in $\S 22.3$ of [my commutative algebra notes](http://alpha.math.uga.edu/%7Epete/integral.pdf).


This paper has spawned a lot of research by ring theorists on **half-factorial domains**: these are rings in which every nonzero nonunit factors into irreducibles and such that the number of irreducible factors is independent of the factorization.


To be honest though, I think there are plenty of number theorists who think of the class number as measuring the failure of unique factorization who don't know Carlitz's theorem (or who know it but are not thinking of it when they make that kind of statement).


Here is another try [**edit: this is essentially the same as Olivier's response, but said differently; I think it is worthwhile to have both**]: when trying to solve certain Diophantine problems (over the integers), one often gets nice results if the class number of a certain number field is prime to a certain quantity. The most famous example of this is Fermat's Last Theorem, which is easy to prove for an odd prime $p$ for which the class number of $\mathbb{Q}(\zeta\_p)$ is prime to $p$: a so-called "regular" prime.


For an application to Mordell equations $y^2 + k = x^3$, see


[http://alpha.math.uga.edu/~pete/4400MordellEquation.pdf](http://alpha.math.uga.edu/%7Epete/4400MordellEquation.pdf)


Especially see Section 4, where the class of rings "of class number prime to 3" is defined axiomatically and applied to the Mordell equation. (N.B.: These notes are written for an advanced undergraduate / first year grad student audience.)


The Mordell equation is probably a better example than the Fermat equation because:


(i) the argument in the "regular" case is more elementary than FLT in the regular case (the latter is too involved to be done in a first course), and


(ii) when the "regularity" hypothesis is dropped, it is not just harder to prove that there are no nontrivial solutions, it is actually very often false!


