nLab algebraic integer




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.


Colloquially, an algebraic integer is a solution to an equation

x n+a 1x n1++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 \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 \mathbb{Q} (e.g., if kk is a number field), an element αk\alpha \in k is integral if the subring [α]k\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 AEA \subseteq E and a finite field extension EFE \subseteq F, an element αF\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 \mathbb{Z} (say), then α+β\alpha + \beta and αβ\alpha \cdot \beta are integral over \mathbb{Z}. For, if β\beta is integral over \mathbb{Z}, it is a fortiori integral over [α]\mathbb{Z}[\alpha], hence

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

is finitely generated over [α]\mathbb{Z}[\alpha] and therefore, since α\alpha is integral, also finitely generated over \mathbb{Z}. It follows that the submodules [α+β]\mathbb{Z}[\alpha + \beta] and [αβ]\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 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.



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Last revised on December 20, 2021 at 17:14:15. See the history of this page for a list of all contributions to it.