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Ring of integers
From Wikipedia, the free encyclopedia
In mathematics, the ring of integers is the set of integers made an algebraic structure Z with the operations of integer addition, negation, and multiplication. It is a commutative ring, and is the prototypical such by virtue of satisfying only those equations holding of all commutative rings with identity; indeed it is the initial commutative ring, as well as being the initial ring.
More generally 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 as above 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.
An alternative term is maximal order, since the ring of integers of a number field is indeed the unique maximal order in the field.
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-2).
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).