Ring of integers

Examples

Computational tool

A useful tool for computing the integral closure of the ring of integers in an algebraic field ${\displaystyle K/\mathbb{Q}}$ is the discriminant. If K is of degree n over ${\displaystyle \mathbb{Q}}$, and ${\displaystyle {\alpha }_1,\dots ,{\alpha }_n\in {\mathcal{O}}_K}$ form a basis of ${\displaystyle K}$ over ${\displaystyle \mathbb{Q}}$, set ${\displaystyle d={\mathrm{\Delta }}_{K/\mathbb{Q}}({\alpha }_1,\dots ,{\alpha }_n)}$. Then, ${\displaystyle {\mathcal{O}}_K}$ is a submodule of the ${\displaystyle \mathbb{Z}}$-module spanned by ${\displaystyle {\alpha }_1/d,\dots ,{\alpha }_n/d}$. ^{pg. 33} In fact, if d is square-free, then ${\displaystyle {\alpha }_1,\dots ,{\alpha }_n}$ forms an integral basis for ${\displaystyle {\mathcal{O}}_K}$. ^{pg. 35}

Cyclotomic extensions

If p is a prime, ζ is a pth root of unity and ${\displaystyle K=\mathbb{Q}(\zeta )}$ is the corresponding cyclotomic field, then an integral basis of ${\displaystyle {\mathcal{O}}_K=\mathbb{Z}[\zeta ]}$ is given by (1, ζ, ζ ², ..., ζ ^p−2).

Quadratic extensions

If ${\displaystyle d}$ is a square-free integer and ${\displaystyle K=\mathbb{Q}(\sqrt{d})}$ is the corresponding quadratic field, then ${\displaystyle {\mathcal{O}}_K}$ is a ring of quadratic integers and its integral basis is given by ${\displaystyle \left(1,\frac{1+\sqrt{d}}{2}\right)}$ if d ≡ 1 (mod 4) and by ${\displaystyle (1,\sqrt{d})}$ if d ≡ 2, 3 (mod 4). This can be found by computing the minimal polynomial of an arbitrary element ${\displaystyle a+b\sqrt{d}\in \mathbb{Q}(\sqrt{d})}$ where ${\displaystyle a,b\in \mathbb{Q}}$.

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 ${\displaystyle \mathbb{Z}[\sqrt{-5}]}$, the element 6 has two essentially different factorizations into irreducibles:

${\displaystyle 6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5}).}$

A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals.

The units of a ring of integers O_K form a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of K. A set of torsion-free generators is called a set of fundamental units.

Generalization

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. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.

For example, the p-adic integers ${\displaystyle {\mathbb{Z}}_p}$ are the ring of integers of the p-adic numbers ${\displaystyle {\mathbb{Q}}_p}$.