In mathematics, the ring of integers of an algebraic number field ${\displaystyle K}$ is the ring of all algebraic integers contained in ${\displaystyle K}$. An algebraic integer is a root of a monic polynomial with integer coefficients: ${\displaystyle x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}}$. This ring is often denoted by ${\displaystyle O_{K}}$ or ${\displaystyle {\mathcal {O}}_{K}}$. Since any integer belongs to ${\displaystyle K}$ and is an integral element of ${\displaystyle K}$, the ring ${\displaystyle \mathbb {Z} }$ is always a subring of ${\displaystyle O_{K}}$.

The ring of integers ${\displaystyle \mathbb {Z} }$ is the simplest possible ring of integers. Namely, ${\displaystyle \mathbb {Z} =O_{\mathbb {Q} }}$ where ${\displaystyle \mathbb {Q} }$ is the field of rational numbers. And indeed, in algebraic number theory the elements of ${\displaystyle \mathbb {Z} }$ are often called the "rational integers" because of this.

The next simplest example is the ring of Gaussian integers ${\displaystyle \mathbb {Z} [i]}$, consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field ${\displaystyle \mathbb {Q} (i)}$ of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, ${\displaystyle \mathbb {Z} [i]}$ 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.

The ring of integers O_K is a finitely-generated ${\displaystyle \mathbb {Z} }$-module. Indeed, it is a free ${\displaystyle \mathbb {Z} }$-module, and thus has an integral basis, that is a basis b₁, ..., b_n ∈ O_K of the ${\displaystyle \mathbb {Q} }$-vector space K such that each element x in O_K can be uniquely represented as

with ${\displaystyle a_{i}\in \mathbb {Z} }$. The rank n of O_K as a free ${\displaystyle \mathbb {Z} }$-module is equal to the degree of K over ${\displaystyle \mathbb {Q} }$.

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 ${\displaystyle \alpha _{1},\ldots ,\alpha _{n}\in {\mathcal {O}}_{K}}$ form a basis of ${\displaystyle K}$ over ${\displaystyle \mathbb {Q} }$, set ${\displaystyle d=\Delta _{K/\mathbb {Q} }(\alpha _{1},\ldots ,\alpha _{n})}$. Then, ${\displaystyle {\mathcal {O}}_{K}}$ is a submodule of the ${\displaystyle \mathbb {Z} }$-module spanned by ${\displaystyle \alpha _{1}/d,\ldots ,\alpha _{n}/d}$. ^{pg. 33} In fact, if d is square-free, then ${\displaystyle \alpha _{1},\ldots ,\alpha _{n}}$ forms an integral basis for ${\displaystyle {\mathcal {O}}_{K}}$. ^{pg. 35}

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, ζ², ..., ζ^p−2).