Ideal class group on arvau

Wikipedia / Ideal class group

In mathematics, the ideal class group (or class group) of an algebraic number field is the quotient group is the group of fractional ideals of the ring of integers is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization. The order of the group, which is finite, is called the class number.

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.

History and origin of the ideal class group

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-torsion in that group for the field-roots of unity, for any prime number, 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.

Definition

is an integral domain, define a relation on nonzero fractional ideals. It is easily shown that this is an equivalence relation. The equivalence classes are called the ideal classes is well-defined and commutative which serves as an identity element has an inverse may only be a monoid.

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) is a product of prime ideals.

Properties

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).

The number of ideal classes—the —may be infinite in general. In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain. is a ring of algebraic integers, then the class number is always finite. This is one of the main results of classical algebraic number theory.

Computation of the class group is hard, in general; it can be done by hand for the ring of integers in an algebraic number field of small discriminant, using Minkowski's bound. This result gives a bound, depending on the ring, such that every ideal class contains an ideal norm less than the bound. In general the bound is not sharp enough to make the calculation practical for fields with large discriminant, but computers are well suited to the task.

to their corresponding class groups is functorial, and the class group can be subsumed under the heading of algebraic K-theory-groups can also be employed and interpreted arithmetically in connection to rings of integers.

Relation with the group of units

It was remarked above that the ideal class group provides part of the answer to the question of how much ideals in a Dedekind domain behave like elements. The other part of the answer is provided by the group of units of the Dedekind domain, since passage from principal ideals to their generators requires the use of units (and this is the rest of the reason for introducing the concept of fractional ideal, as well).

by sending every element to the fractional ideal it generates. This is a group homomorphism; its kernel, and its cokernel. The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.

Examples of ideal class groups

, respectively the integers, Gaussian integers, and Eisenstein integers, are all principal ideal domains (and in fact are all Euclidean domains), and so have class number 1: i.e., they have trivial ideal class groups.
is a field, then the polynomial ring is an integral domain. It has a countably infinite set of ideal classes.

Class numbers of quadratic fields

is a square-free integer is a quadratic extension. This result was first conjectured by Gauss and proven by Kurt Heegner, although Heegner's proof was not believed until Harold Stark gave a later proof in 1967 (see Stark–Heegner theorem). This is a special case of the famous class number problem.

with class number 1. Computational results indicate that there are a great many such fields. However, it is not even known if there are infinitely many number fields with class number 1.

is isomorphic to the class group of integral binary quadratic forms of discriminant, the ideal class group may be half the size since the class group of integral binary quadratic forms is isomorphic to the narrow class group.

For real quadratic integer rings, the class number is given in OEIS A003649; for the imaginary case, they are given in OEIS A000924.

Example of a non-trivial class group

is cyclic of order 2. Indeed, the ideal

is not principal, which can be proved by contradiction has a multiplicative norm.

, because the quotient ring is isomorphic has no elements of norm 2, because the Diophantine equation has no solutions in integers, as it has no solutions modulo 5.

in the ideal class group has order two. Showing that there aren't any other ideal classes requires more effort.

has two distinct factorisations into irreducibles:

Connections to class field theory

Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group. A particularly beautiful example is found in the Hilbert class field of a number field, which can be defined as the maximal unramified abelian extension of such a field. The Hilbert class field L of a number field K is unique and has the following properties:

Every ideal of the ring of integers of K becomes principal in L, i.e., if I is an integral ideal of K then the image of I is a principal ideal in L.
L is a Galois extension of K with Galois group isomorphic to the ideal class group of K.

Neither property is particularly easy to prove.