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Concept
Ring of integers
Summary
In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of , the ring is always a subring of .
The ring of integers 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 next simplest example is the ring of Gaussian integers , consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, 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.
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.
A useful tool for computing the integral closure of the ring of integers in an algebraic field K/Q is the discriminant. If K is of degree n over Q, and form a basis of K over Q, set . Then, is a submodule of the Z-module spanned by . pg. 33 In fact, if d is square-free, then forms an integral basis for . pg. 35
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[ζ] is given by (1, ζ, ζ 2, ..., ζ p−2).
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 (1, (1 + ) /2) if d ≡ 1 (mod 4) and by (1, ) if d ≡ 2, 3 (mod 4).
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