In mathematics, the ring of integers of an algebraic number field (also sometimes called the number ring corresponding to number field ) is the ring of all algebraic integers. An algebraic integer is a root of a monic polynomial with integer coefficients.. Since any integer and is an integral element is always a subring.

is the simplest possible ring of integers. is the field of rational numbers. And indeed, in algebraic number theory are often called the "rational integers" because of this.

The next simplest example is the ring of Gaussian integers, consisting of complex numbers whose real and imaginary parts of Gaussian rationals is a Euclidean domain.

The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.

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 -vector space such that each element in can be uniquely represented as

. The rank of -module is equal to the degree of.

Examples

Computational tool

is the discriminant. If is of degree is a submodule of the . pg. 33 In fact, if . pg. 35

Cyclotomic extensions

If is a prime, is a th root of unity is the corresponding cyclotomic field is given by .

Quadratic extensions

is a square-free integer is the corresponding quadratic field is a ring of quadratic integers if if . This can be found by computing the minimal polynomial.

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, 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 is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of . 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 p-adic integers are the ring of integers of the p-adic numbers.

See also

• Minimal polynomial (field theory)

• Integral closure – gives a technique for computing integral closures

Notes

Citations

References