Ring of integers

The ring of integers is a finitely-generated \mathbb Z-module. Indeed, it is a free \mathbb Z-module, and thus has an integral basis, that is a basis of the \mathbb Q-vector space  such that each element  in can be uniquely represented as :x=\sum_{i=1}^na_ib_i, with a_i \in \mathbb Z. The rank  of as a free \mathbb Z-module is equal to the degree of  over \mathbb Q. ==Examples==

Computational tool A useful tool for computing the integral closure of the ring of integers in an algebraic field K / \mathbb Q is the discriminant. If is of degree over \mathbb Q, and \alpha_1,\ldots,\alpha_n \in \mathcal{O}_K form a basis of K over \mathbb Q, set d = \Delta_{K/\mathbb{Q}}(\alpha_1,\ldots,\alpha_n). Then, \mathcal{O}_K is a submodule of the spanned by \alpha_1/d,\ldots,\alpha_n/d. pg. 33 In fact, if is square-free, then \alpha_1,\ldots,\alpha_n forms an integral basis for \mathcal{O}_K. pg. 35 Cyclotomic extensions If is a prime,  is a th root of unity and K = \mathbb Q(\zeta) is the corresponding cyclotomic field, then an integral basis of \mathcal{O}_K=\mathbb{Z}[\zeta] is given by . Quadratic extensions If d is a square-free integer and K = \mathbb{Q}(\sqrt{d}\,) is the corresponding quadratic field, then \mathcal{O}_K is a ring of quadratic integers and its integral basis is given by \left(1, \frac{1 + \sqrt{d}}{2}\right) if and by (1, \sqrt{d}) if . This can be found by computing the minimal polynomial of an arbitrary element a + b\sqrt{d} \in \mathbb{Q}(\sqrt{d}) where a,b \in \mathbb{Q}. ==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 \mathbb Z[\sqrt{-5}], the element 6 has two essentially different factorizations into irreducibles: : 6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}). 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 \mathbb Z_p are the ring of integers of the -adic numbers \mathbb Q_p. == See also ==