Ideal class group

In mathematics, the ideal class group (or class group) of an algebraic number field ${\displaystyle K}$ is the quotient group ${\displaystyle J_{K}/P_{K}}$ where ${\displaystyle J_{K}}$ is the group of fractional ideals of the ring of integers of ${\displaystyle K}$, and ${\displaystyle P_{K}}$ is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of ${\displaystyle K}$. The order of the group, which is finite, is called the class number of ${\displaystyle K}$.

The theory extends to Dedekind domains and their fields of fractions, for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain.

Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an ideal was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form by Carl Friedrich Gauss, a composition law was defined on certain equivalence classes of forms. This gave a finite abelian group, as was recognised at the time.

Later Ernst Kummer was working towards a theory of cyclotomic fields. It had been realised (probably by several people) that failure to complete proofs in the general case of Fermat's Last Theorem by factorisation using the roots of unity was for a very good reason: a failure of unique factorization – i.e., the fundamental theorem of arithmetic – to hold in the rings generated by those roots of unity was a major obstacle. Out of Kummer's work for the first time came a study of the obstruction to the factorization. We now recognise this as part of the ideal class group: in fact Kummer had isolated the ${\displaystyle p}$-torsion in that group for the field of ${\displaystyle p}$-roots of unity, for any prime number ${\displaystyle p}$, as the reason for the failure of the standard method of attack on the Fermat problem (see regular prime).

Somewhat later again Richard Dedekind formulated the concept of an ideal, Kummer having worked in a different way. At this point the existing examples could be unified. It was shown that while rings of algebraic integers do not always have unique factorization into primes (because they need not be principal ideal domains), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals (that is, every ring of algebraic integers is a Dedekind domain). The size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal ideal domain; a ring is a principal ideal domain if and only if it has a trivial ideal class group.

If ${\displaystyle R}$ is an integral domain, define a relation ${\displaystyle \sim }$ on nonzero fractional ideals of ${\displaystyle R}$ by ${\displaystyle I\sim J}$ whenever there exist nonzero elements ${\displaystyle a}$ and ${\displaystyle b}$ of ${\displaystyle R}$ such that ${\displaystyle (a)I=(b)J}$. It is easily shown that this is an equivalence relation. The equivalence classes are called the ideal classes of ${\displaystyle R}$. Ideal classes can be multiplied: if ${\displaystyle [I]}$ denotes the equivalence class of the ideal ${\displaystyle I}$, then the multiplication ${\displaystyle [I][J]=[IJ]}$ is well-defined and commutative. The principal ideals form the ideal class ${\displaystyle [R]}$ which serves as an identity element for this multiplication. Thus a class ${\displaystyle [I]}$ has an inverse ${\displaystyle [J]}$ if and only if there is an ideal ${\displaystyle J}$ such that ${\displaystyle IJ}$ is a principal ideal. In general, such a ${\displaystyle J}$ may not exist and consequently the set of ideal classes of ${\displaystyle R}$ may only be a monoid.

However, if ${\displaystyle R}$ is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of ${\displaystyle R}$. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal (except ${\displaystyle R}$) is a product of prime ideals.

The ideal class group is trivial (i.e. has only one element) if and only if all ideals of ${\displaystyle R}$ are principal. In this sense, the ideal class group measures how far ${\displaystyle R}$ is from being a principal ideal domain, and hence from satisfying unique prime factorization (Dedekind domains are unique factorization domains if and only if they are principal ideal domains).