symmetric monoidal (∞,1)-category of spectra
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
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
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.
The algebraic integers in the rational numbers are the ordinary integers.
The algebraic integers in the Gaussian numbers are the Gaussian integers.
Last revised on October 5, 2018 at 08:51:38. See the history of this page for a list of all contributions to it.