Ring of integers
Template:Short description Template:Ring theory sidebar In mathematics, the ring of integers of an algebraic number field <math>K</math> is the ring of all algebraic integers contained in <math>K</math>.Template:Sfn An algebraic integer is a root of a monic polynomial with integer coefficients: <math>x^n+c_{n-1}x^{n-1}+\cdots+c_0</math>.Template:Sfn This ring is often denoted by <math>O_K</math> or <math>\mathcal O_K</math>. Since any integer belongs to <math>K</math> and is an integral element of <math>K</math>, the ring <math>\mathbb{Z}</math> is always a subring of <math>O_K</math>.
The ring of integers <math>\mathbb{Z}</math> is the simplest possible ring of integers.Template:Efn Namely, <math>\mathbb{Z}=O_{\mathbb{Q}}</math> where <math>\mathbb{Q}</math> is the field of rational numbers.Template:Sfn And indeed, in algebraic number theory the elements of <math>\mathbb{Z}</math> are often called the "rational integers" because of this.
The next simplest example is the ring of Gaussian integers <math>\mathbb{Z}[i]</math>, consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field <math>\mathbb{Q}(i)</math> of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, <math>\mathbb{Z}[i]</math> 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.Template:Sfn
Properties
The ring of integers Template:Math is a finitely-generated <math>\mathbb Z</math>-module. Indeed, it is a free <math>\mathbb Z</math>-module, and thus has an integral basis, that is a basis Template:Math of the <math>\mathbb Q</math>-vector space Template:Mvar such that each element Template:Mvar in Template:Math can be uniquely represented as
- <math>x=\sum_{i=1}^na_ib_i,</math>
with <math>a_i \in \mathbb Z</math>.<ref name=Cas193>Cassels (1986) p. 193</ref> The rank Template:Mvar of Template:Math as a free <math>\mathbb Z</math>-module is equal to the degree of Template:Mvar over <math>\mathbb Q</math>.
Examples
Computational tool
A useful tool for computing the integral closure of the ring of integers in an algebraic field <math>K / \mathbb Q</math> is the discriminant. If Template:Math is of degree Template:Math over <math>\mathbb Q</math>, and <math>\alpha_1,\ldots,\alpha_n \in \mathcal{O}_K</math> form a basis of <math>K</math> over <math>\mathbb Q</math>, set <math>d = \Delta_{K/\mathbb{Q}}(\alpha_1,\ldots,\alpha_n)</math>. Then, <math>\mathcal{O}_K</math> is a submodule of the Template:Nowrap spanned by <math>\alpha_1/d,\ldots,\alpha_n/d</math>.<ref name=":0">Template:Cite web</ref> pg. 33 In fact, if Template:Math is square-free, then <math>\alpha_1,\ldots,\alpha_n</math> forms an integral basis for <math>\mathcal{O}_K</math>.<ref name=":0" /> pg. 35
Cyclotomic extensions
If Template:Mvar is a prime, Template:Math is a Template:Mvarth root of unity and <math>K = \mathbb Q(\zeta)</math> is the corresponding cyclotomic field, then an integral basis of <math>\mathcal{O}_K=\mathbb{Z}[\zeta]</math> is given by Template:Math.Template:Sfn
Quadratic extensions
If <math>d</math> is a square-free integer and <math>K = \mathbb{Q}(\sqrt{d}\,)</math> is the corresponding quadratic field, then <math>\mathcal{O}_K</math> is a ring of quadratic integers and its integral basis is given by <math>\left(1, \frac{1 + \sqrt{d}}{2}\right)</math> if Template:Math and by <math>(1, \sqrt{d})</math> if Template:Math.Template:Sfn This can be found by computing the minimal polynomial of an arbitrary element <math>a + b\sqrt{d} \in \mathbb{Q}(\sqrt{d})</math> where <math>a,b \in \mathbb{Q}</math>.
Multiplicative structure
In a ring of integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorization: for example, in the ring of integers <math>\mathbb Z[\sqrt{-5}]</math>, the element 6 has two essentially different factorizations into irreducibles:Template:Sfn<ref>Template:Cite book</ref>
- <math> 6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}).</math>
A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals.Template:Sfn
The units of a ring of integers Template:Math is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of Template:Math. A set of torsion-free generators is called a set of fundamental units.Template:Sfn
Generalization
One defines the ring of integers of a non-archimedean local field Template:Math as the set of all elements of Template:Math with absolute value Template:Math; this is a ring because of the strong triangle inequality.Template:Sfn If Template:Math is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.Template:Sfn
For example, the [[p-adic integer|Template:Mvar-adic integers]] <math>\mathbb Z_p</math> are the ring of integers of the [[p-adic number|Template:Mvar-adic numbers]] <math>\mathbb Q_p</math>.
See also
- Minimal polynomial (field theory)
- Integral closure – gives a technique for computing integral closures
Notes
Citations