Class group

Class group in the context of algebraic number theory is a fundamental concept that plays a crucial role in the study of number fields and their ring of integers. The class group measures the failure of unique factorization in the ring of integers of a number field. It is a major object of study in both classical and modern number theory, providing insights into the structure and properties of number fields.

Definition

The class group of a number field K is defined in terms of its ring of integers O_K. A ring of integers is the set of all algebraic integers in K, which are roots of monic polynomials with coefficients in the integers. The class group, denoted as Cl(K) or sometimes Pic(O_K), is the group of fractional ideal classes of O_K. A fractional ideal is a finitely generated O_K-submodule of K that contains a non-zero element of O_K for which its inverse is also in the module, making the module a fractional ideal. The class group is the quotient of the group of all non-zero fractional ideals by the subgroup of principal ideals (ideals generated by a single element).

Importance

The class group encapsulates information about the extent to which unique factorization fails in O_K. If Cl(K) is trivial (i.e., consists only of the identity element), then O_K is a unique factorization domain (UFD), meaning every element of O_K can be uniquely factored into prime elements up to order and units. The size of the class group, known as the class number, therefore, provides a measure of the "distance" from O_K being a UFD.

Computation

Computing the class group of a given number field is a significant challenge in computational number theory. Various algorithms exist for this purpose, with the most efficient ones relying on techniques from lattice basis reduction, such as the LLL algorithm, and the use of elliptic curves.

Examples

A simple example is the ring of integers of the quadratic number field Q(√d), where d is a square-free integer. The structure of the class group in this case can often be determined explicitly and is related to the continued fraction expansion of √d.

Applications

Class groups have applications in various areas of mathematics and computer science, including cryptographic algorithms that rely on the hardness of computing class groups of certain number fields. They also appear in the study of L-functions and modular forms, playing a role in the proof of the Fermat's Last Theorem and the Birch and Swinnerton-Dyer conjecture.

See Also

• Algebraic number theory
• Ring of integers
• Unique factorization domain
• Computational number theory
• Lattice basis reduction