Ring of integers

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In mathematics, the phrase ring of integers can refer to

• the ring of all integers (whole numbers: positive, negative or zero), usually denoted by Z.
• given an algebraic number field K, the algebraic integers contained in K form a ring, the ring of integers of K, often denoted by O_K.

Using this notation, we can write Z = O_Q 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 O_K has an integral basis; by this we mean that there exist b₁,...,b_n ∈ O_K (the integral basis) such that each element x in O_K can uniquely be represented as

with a_i ∈ Z.

If ζ is a pth root of unity and K=Q(ζ) is the corresponding cyclotomic field, then an integral basis of O_K is given by (1,ζ,ζ²,...,ζ^p-2).

If d is a square-free integer and K=Q(d^1/2) is the corresponding quadratic field, then an integral basis of O_K is given by (1,(1+d^1/2)/2) if d≡1 (mod 4) and by (1,d^1/2) if d≡2 or 3 (mod 4).