Ring of integers
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In mathematics, the ring of integers of an algebraic number field K, often denoted by OK, is the ring of algebraic integers contained in K.
Using this notation, we can write Z = OQ since Z is the ring of integers of the field Q 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 OK has an integral basis; by this we mean that there exist b1,...,bn ∈ OK (the integral basis) such that each element x in OK can uniquely be represented as
with ai ∈ Z.
If ζ is a pth root of unity and K=Q(ζ) is the corresponding cyclotomic field, then an integral basis of OK is given by (1,ζ,ζ2,...,ζp-1).
If d is a square-free integer and K=Q(d1/2) is the corresponding quadratic field, then an integral basis of OK is given by (1,(1+d1/2)/2) if d≡1 (mod 4) and by (1,d1/2) if d≡2 or 3 (mod 4).
