nLab algebraic integer

Idea

In number theory, the concept of algebraic integer is a generalization of that of integer to more general base-number fields. These algebraic integers form what is called the ring of integers and so in order to distinguish that from the standard integers $\mathbb{Z}$ these are sometimes called rational integers, since they are the algebraic integers in the ring of rational numbers.

Definition

Colloquially, an algebraic integer is a solution to an equation

$$x^n + a_1 x^{n-1} + \ldots + a_n = 0 \qquad (1)$$

where each $a_i$ is an integer (hence a root of the polynomial on the left). More precisely, an element $x$ belonging to an algebraic extension of the rational numbers $\mathbb{Q}$ is an (algebraic) integer, or more briefly is integral, if it satisfies an equation of the form (1). Equivalently, if $k$ is an algebraic extension of $\mathbb{Q}$ (e.g., if $k$ is a number field), an element $\alpha \in k$ is integral if the subring $\mathbb{Z}[\alpha] \subseteq k$ is finitely generated as a $\mathbb{Z}$-module.

This notion may be relativized as follows: given an integral domain in its field of fractions $A \subseteq E$ and a finite field extension $E \subseteq F$, an element $\alpha \in F$ is integral over $A$ if $A[\alpha] \subseteq F$ is finitely generated as an $A$-module.

If $\alpha, \beta$ are integral over $\mathbb{Z}$ (say), then $\alpha + \beta$ and $\alpha \cdot \beta$ are integral over $\mathbb{Z}$. For, if $\beta$ is integral over $\mathbb{Z}$, it is a fortiori integral over $\mathbb{Z}[\alpha]$, hence

$$(\mathbb{Z}[\alpha])[\beta] = \mathbb{Z}[\alpha, \beta]$$

is finitely generated over $\mathbb{Z}[\alpha]$ and therefore, since $\alpha$ is integral, also finitely generated over $\mathbb{Z}$. It follows that the submodules $\mathbb{Z}[\alpha + \beta]$ and $\mathbb{Z}[\alpha \cdot \beta]$ are therefore also finitely generated over $\mathbb{Z}$ (since $\mathbb{Z}$ is a Noetherian ring). Thus the integral elements form a ring. In particular, the integral elements in a number field $k$ form a ring often denoted by $\mathcal{O}_k$, usually called the ring of integers in $k$. This ring is a Dedekind domain.

Examples

• The algebraic integers in the rational numbers are the ordinary integers.

• The algebraic integers in the Gaussian numbers are the Gaussian integers.

• golden ratio

Related concepts

• ring of integers

References

Textbook account:

• J. W. S. Cassels, Section 10.3 of: Local Fields, Cambridge University Press, 1986 (ISBN:9781139171885, doi:10.1017/CBO9781139171885)

Lecture notes:

• James Milne, Chapter 2 of: Algebraic number theory, 2020 (pdf)