Ring of integers

In mathematics, the ring of integers of an algebraic number field K is the ring of all integral elements contained in K. An integral element is a root of a monic polynomial with integer coefficients, xn + cn−1xn−1 + … + c0 . This ring is often denoted by OK or O K {displaystyle {mathcal {O}}_{K}} . Since any integer number belongs to K and is an integral element of K, the ring Z is always a subring of OK. In mathematics, the ring of integers of an algebraic number field K is the ring of all integral elements contained in K. An integral element is a root of a monic polynomial with integer coefficients, xn + cn−1xn−1 + … + c0 . This ring is often denoted by OK or O K {displaystyle {mathcal {O}}_{K}} . Since any integer number belongs to K and is an integral element of K, the ring Z is always a subring of OK. The ring Z is the simplest possible ring of integers. Namely, Z = OQ where Q is the field of rational numbers. And indeed, in algebraic number theory the elements of Z 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. The ring of integers OK is a finitely-generated Z-module. Indeed, it is a free Z-module, and thus has an integral basis, that is a basis b1, … ,bn ∈ OK of the Q-vector space K such that each element x in OK can be uniquely represented as with ai ∈ Z. The rank n of OK as a free Z-module is equal to the degree of K over Q. The rings of integers in number fields are Dedekind domains. If p is a prime, ζ is a pth root of unity and K = Q(ζ) is the corresponding cyclotomic field, then an integral basis of OK = Z is given by (1, ζ, ζ2, … , ζp−2). If d is a square-free integer and K = Q(√d) is the corresponding quadratic field, then OK is a ring of quadratic integers and its integral basis is given by (1, (1 + √d)/2) if d ≡ 1 (mod 4) and by (1, √d) if d ≡ 2, 3 (mod 4). In a ring of integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorisation: for example, in the ring of integers ℤ, the element 6 has two essentially different factorisations into irreducibles: