Ring of integers

In mathematics, the ring of integers of an algebraic number field ${\displaystyle K}$ is the ring of all algebraic integers contained in ${\displaystyle K}$.^[1] An algebraic integer is a root of a monic polynomial with integer coefficients: ${\displaystyle x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}}$.^[2] 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.^[a] Namely, ${\displaystyle \mathbb {Z} =O_{\mathbb {Q} }}$ where ${\displaystyle \mathbb {Q} }$ is the field of rational numbers.^[3] 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.^[4]

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