# Ring of integers

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

The ring of integers $\mathbb {Z}$ is the simplest possible ring of integers.[lower-alpha 1] Namely, $\mathbb {Z} =O_{\mathbb {Q} }$ where $\mathbb {Q}$ is the field of rational numbers. And indeed, in algebraic number theory the elements of $\mathbb {Z}$ are often called the "rational integers" because of this.

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