algebraic integer in nLab

Contents

Contents

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 ℤ\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 1x n−1+…+a n=0(1)x^n + a_1 x^{n-1} + \ldots + a_n = 0 \qquad (1)

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

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

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

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

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

• 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)