Ring of integers
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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 rational integer coefficients, xn + cn−1xn−1 + … + c0 . This ring is often denoted by OK or . Since any integer number belongs to K and is its integral element, 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 a number field is the unique maximal order in the field.
The ring of integers OK is a 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 d is a square-free integer and K = Q(√) is the corresponding quadratic field, then OK 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).
One defines the ring of integers of a non-archimedean local field F as the set of all elements of F with absolute value ≤1; this is a ring because of the strong triangle inequality. If F is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers.
For example, the p-adic integers Zp are the ring of integers of the p-adic numbers Qp .
- Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
- The ring of integers, without specifying the field, refers to the ring Z of "ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "integer" in abstract algebra.