nLab ring of integers

Idea

Numbers like $\sqrt{3}$ and ${}^5\sqrt{3}$ look tied to the usual integers: indeed they are obtained from an integer number by doing operation of 2nd or 5th root. More generally, we can start with the usual integer numbers and do similar algebraic operations, namely form monic polynomials with such integer coefficients and search for roots in a larger ring, obtaining generalized integers – called algebraic integers – as solutions of a monic polynomial equation, which makes sense in an arbitrary commutative ring.

If we look for solutions of monic equations within $\mathbb{Q}$, the field of rationals, we get nothing new, just the usual integers. Thus the integral elements in a ring generalize integers in wide context and form a “ring of integers” within a larger ring. This generalizes the standard integers inside the field of rational numbers to various situations like number fields and local non-archimedean fields.

Definition

General definition

In any unital ring $R$ one can identify the ring $\mathbb{Z}$ with the subring $\{ n 1_R | n\in\mathbb{Z}\}\subseteq R$ of all multiples $\pm 1_R, \pm (1_R+1_R), \pm(1_R+1_R+1_R),\ldots$ of the unit element $1_R$. The ring $R$ is then a left and right module over $\mathbb{Z}$ via the multiplication with the corresponding multiple of unit, that is $n.m = (n 1_R)\cdot m$.

An algebraic integer in a commutative ring $R$ is any element $r \in R$ which satisfies equation $P(r) = 0$ where $P$ is a nontrivial polynomial whose coefficients are multiplies of $1_R$ and the top degree coefficient is $1_R$ (in other words, a root of a monic polynomial in $R$ with the coefficients in $\mathbb{Z}$).

It can be checked that the set of algebraic integers (also said to be integral elements) in $R$ (also said the ring of integers of $R$) is closed with respect to addition, multiplication and taking the negative of an element, hence a subring of $R$, which is moreover containing the usual integers $\mathbb{Z} 1_R$ as unique solutions for $x\in R$ of equations $1_R \cdot x - n 1_R = 0$.

Notation in number theory

The subring of integers in an algebraic number field $K$ (a finite-dimensional field extension of $\mathbb{Q}$), is often denoted $\mathcal{O}_K \hookrightarrow K$. In a local non-archimedean field $F$, then its ring of integers is often denoted $\mathcal{O}_F$.

Properties in a number field

Alternatively, an element $\alpha \in K$ is an algebraic integer if the subring $\mathbb{Z}[\alpha] \hookrightarrow K$ generated by $\alpha$ is of finite rank over $\mathbb{Z}$. This makes it plain that the algebraic integers themselves form a ring: if $\alpha, \beta \in \mathcal{O}_K$ are integral, then both $\alpha - \beta$ and $\alpha \beta$ are contained in the image of the induced map $\mathbb{Z}[\alpha] \otimes \mathbb{Z}[\beta] \to K$, which is also of finite rank.

Properties in a local non-archimedean field

Given a local non-archimedean field $F$, then its ring of integers $\mathcal{O}_F$ is the subring of elements of norm $\leq 1$.

If $F$ is the formal completion of a number field $K$, then the ring of integers of $F$ is the formal completion of the ring of integers of $K$.

Examples

• For $K = \mathbb{Q}$ the rational numbers then $\mathcal{O}_{\mathbb{Q}} \simeq \mathbb{Z}$ is the commutative ring of ordinary integers.

• For $p$ any prime and $\mathbb{Q}_p$ the formal completion of $\mathbb{Q}$ at $p$, hence the p-adic numbers, then the ring of integers of $\mathbb{Q}_p$ is $\mathbb{Z}_p$, the p-adic integers.

• For $K$ the Gaussian numbers then $\mathcal{O}_K$ is the ring of Gaussian integers.

• The ring of integers of a cyclotomic field $\mathbb{Q}(\zeta_n)$ is $\mathbb{Z}[\zeta_n]$, called the ring of cyclotomic integers.

Properties

General

A ring of integers is a Dedekind domain.

Function field analogy

function field analogy

	number fields (“function fields of curves over F1”)	function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)	Riemann surfaces/complex curves
affine and projective line
	$\mathbb{Z}$ (integers)	$\mathbb{F}_q[z]$ (polynomials, polynomial algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
	$\mathbb{Q}$ (rational numbers)	$\mathbb{F}_q(z)$ (rational fractions/rational function on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	meromorphic functions on complex plane
	$p$ (prime number/non-archimedean place)	$x \in \mathbb{F}_p$, where $z - x \in \mathbb{F}_q[z]$ is the irreducible monic polynomial of degree one	$x \in \mathbb{C}$, where $z - x \in \mathcal{O}_{\mathbb{C}}$ is the function which subtracts the complex number $x$ from the variable $z$
	$\infty$ (place at infinity)		$\infty$
	$Spec(\mathbb{Z})$ (Spec(Z))	$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)	complex plane
	$Spec(\mathbb{Z}) \cup place_{\infty}$	$\mathbb{P}_{\mathbb{F}_q}$ (projective line)	Riemann sphere
	$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)	$\frac{\partial}{\partial z}$ (coordinate derivation)	“
	genus of the rational numbers = 0		genus of the Riemann sphere = 0
formal neighbourhoods
	$\mathbb{Z}/(p^n \mathbb{Z})$ (prime power local ring)	$\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])$ ($n$-th order univariate local Artinian $\mathbb{F}_q$-algebra)	$\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])$ ($n$-th order univariate Weil $\mathbb{C}$-algebra)
	$\mathbb{Z}_p$ (p-adic integers)	$\mathbb{F}_q[ [ z -x ] ]$ (power series around $x$)	$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
	$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)		formal disks in $X$
	$\mathbb{Q}_p$ (p-adic numbers)	$\mathbb{F}_q((z-x))$ (Laurent series around $x$)	$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
	$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)	$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
	$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)	$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
	Jacobi theta function
zeta functions
	Riemann zeta function	Goss zeta function
branched covering curves
	$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)	$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$	$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
	$\mathcal{O}_K$ (ring of integers)		$\mathcal{O}_{\Sigma}$ (structure sheaf)
	$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)	$\Sigma$ (arithmetic curve)	$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
	$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)	$\frac{\partial}{\partial z}$	“
	genus of a number field	genus of an algebraic curve	genus of a surface
formal neighbourhoods
	$v$ prime ideal in ring of integers $\mathcal{O}_K$	$x \in \Sigma$	$x \in \Sigma$
	$K_v$ (formal completion at $v$)		$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
	$\mathcal{O}_{K_v}$ (ring of integers of formal completion)		$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
	$\mathbb{A}_K$ (ring of adeles)		$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
	$\mathcal{O}$		$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
	$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)		$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
	Galois group	“	$\pi_1(\Sigma)$ fundamental group
	Galois representation	“	flat connection (“local system”) on $\Sigma$
class field theory
	class field theory	“	geometric class field theory
	Hilbert reciprocity law	Artin reciprocity law	Weil reciprocity law
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)	“
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$	“	$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
	number field Langlands correspondence	function field Langlands correspondence	geometric Langlands correspondence
	$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)	“	$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
	Tamagawa-Weil for number fields	Tamagawa-Weil for function fields
theta functions
	Hecke theta function		functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
	Dedekind zeta function	Weil zeta function	zeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functions	Hasse-Weil zeta function

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)

See also:

• Wikipedia, Ring of integers

The following paper shows that the subset of integers is definable in $\mathbb{Q}$ by a universal first-order formula in the language of rings.

• Jochen Koenigsmann, Defining $\mathbb{Z}$ in $\mathbb{Q}$, Annals of Mathematics, 183, issue 1 (2016) pp 73-93, doi:10.4007/annals.2016.183.1.2, arXiv:1011.3424