Ring of integers


In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in . An integral element is a root of a monic polynomial with integer coefficients,. This ring is often denoted by or. Since any integer number belongs to and is an integral element of , the ring is always a subring of .
The ring is the simplest possible ring of integers. Namely, where is the field of rational numbers. And indeed, in algebraic number theory the elements of are often called the "rational integers" because of this.
The ring of integers of an algebraic number field is the unique maximal order in the field.

Properties

The ring of integers is a finitely-generated -module. Indeed, it is a free -module, and thus has an integral basis, that is a basis of the -vector space such that each element in can be uniquely represented as
with. The rank of as a free -module is equal to the degree of over.
The rings of integers in number fields are Dedekind domains.

Examples

Computational tool

A useful tool for computing the integral close of the ring of integers in an algebraic field is using the discriminant. If is of degree over, and form a basis of over, set. Then, is a submodule of the module spanned by pg. 33. In fact, if is square-free, then this forms an integral basis for pg. 35.

Cyclotomic extensions

If is a prime, ζ is a th root of unity and is the corresponding cyclotomic field, then an integral basis of is given by.

Quadratic extensions

If is a square-free integer and is the corresponding quadratic field, then is a ring of quadratic integers and its integral basis is given by if and by if. This can be found by computing the minimal polynomial of an arbitrary element where.

Multiplicative structure

In a ring of integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorization: for example, in the ring of integers ℤ, the element 6 has two essentially different factorizations into irreducibles:
A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals.
The units of a ring of integers OK is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of K. A set of torsion-free generators is called a set of fundamental units.

Generalization

One defines the ring of integers of a non-archimedean local field as the set of all elements of with absolute value ; this is a ring because of the strong triangle inequality. If is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.
For example, the -adic integers are the ring of integers of the -adic numbers.