In mathematics, the ring of integers of an algebraic number field <math>K</math> (also sometimes called the number ring corresponding to number field <math>K</math>) is the ring of all algebraic integers contained in <math>K</math>. An algebraic integer is a root of a monic polynomial with integer coefficients: <math>x^n+c_{n-1}x^{n-1}+\cdots+c_0</math>. This ring is often denoted by <math>O_K</math> or <math>\mathcal O_K</math>. Since any integer belongs to <math>K</math> and is an integral element of <math>K</math>, the ring <math>\mathbb{Z}</math> is always a subring of <math>O_K</math>.
The ring of integers <math>\mathbb{Z}</math> is the simplest possible ring of integers. Namely, <math>\mathbb{Z}=O_{\mathbb{Q</math> where <math>\mathbb{Q}</math> is the field of rational numbers. And indeed, in algebraic number theory the elements of <math>\mathbb{Z}</math> are often called the "rational integers" because of this.
The next simplest example is the ring of Gaussian integers <math>\mathbb{Z}[i]</math>, consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field <math>\mathbb{Q}(i)</math> of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, <math>\mathbb{Z}[i]</math> 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 <math>\mathbb Z</math>-module. Indeed, it is a free <math>\mathbb Z</math>-module, and thus has an integral basis, that is a basis of the <math>\mathbb Q</math>-vector space such that each element in can be uniquely represented as
:<math>x=\sum_{i=1}^na_ib_i,</math>
with <math>a_i \in \mathbb Z</math>. The rank of as a free <math>\mathbb Z</math>-module is equal to the degree of over <math>\mathbb Q</math>.
Examples
Computational tool
A useful tool for computing the integral closure of the ring of integers in an algebraic field <math>K / \mathbb Q</math> is the discriminant. If is of degree over <math>\mathbb Q</math>, and <math>\alpha_1,\ldots,\alpha_n \in \mathcal{O}_K</math> form a basis of <math>K</math> over <math>\mathbb Q</math>, set <math>d = \Delta_{K/\mathbb{Q(\alpha_1,\ldots,\alpha_n)</math>. Then, <math>\mathcal{O}_K</math> is a submodule of the spanned by <math>\alpha_1/d,\ldots,\alpha_n/d</math>. <sup>pg. 33</sup> In fact, if is square-free, then <math>\alpha_1,\ldots,\alpha_n</math> forms an integral basis for <math>\mathcal{O}_K</math>.
:<math> 6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}).</math>
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 -adic integers <math>\mathbb Z_p</math> are the ring of integers of the -adic numbers <math>\mathbb Q_p</math>.
See also
- Minimal polynomial (field theory)
- Integral closure – gives a technique for computing integral closures