nLab ring of integers




Numbers like 3\sqrt{3} and 53{}^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.


General definition

In any unital ring RR one can identify the ring \mathbb{Z} with the subring {n1 R|n}R\{ n 1_R | n\in\mathbb{Z}\}\subseteq R of all multiples ±1 R,±(1 R+1 R),±(1 R+1 R+1 R),\pm 1_R, \pm (1_R+1_R), \pm(1_R+1_R+1_R),\ldots of the unit element 1 R1_R. The ring RR is then a left and right module over \mathbb{Z} via the multiplication with the corresponding multiple of unit, that is n.m=(n1 R)mn.m = (n 1_R)\cdot m.

An algebraic integer in a commutative ring RR is any element rRr \in R which satisfies equation P(r)=0P(r) = 0 where PP is a nontrivial polynomial whose coefficients are multiplies of 1 R1_R and the top degree coefficient is 1 R1_R (in other words, a root of a monic polynomial in RR with the coefficients in \mathbb{Z}).

It can be checked that the set of algebraic integers (also said to be integral elements) in RR (also said the ring of integers of RR) is closed with respect to addition, multiplication and taking the negative of an element, hence a subring of RR, which is moreover containing the usual integers 1 R\mathbb{Z} 1_R as unique solutions for xRx\in R of equations 1 Rxn1 R=01_R \cdot x - n 1_R = 0.

Notation in number theory

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

Properties in a number field

Alternatively, an element αK\alpha \in K is an algebraic integer if the subring [α]K\mathbb{Z}[\alpha] \hookrightarrow K generated by α\alpha is of finite rank over \mathbb{Z}. This makes it plain that the algebraic integers themselves form a ring: if α,β𝒪 K\alpha, \beta \in \mathcal{O}_K are integral, then both αβ\alpha - \beta and αβ\alpha \beta are contained in the image of the induced map [α][β]K\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 FF, then its ring of integers 𝒪 F\mathcal{O}_F is the subring of elements of norm 1\leq 1.

If FF is the formal completion of a number field KK, then the ring of integers of FF is the formal completion of the ring of integers of KK.




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 𝔽 q\mathbb{F}_q (arithmetic curves)Riemann surfaces/complex curves
affine and projective line
\mathbb{Z} (integers)𝔽 q[z]\mathbb{F}_q[z] (polynomials, polynomial algebra on affine line 𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q})𝒪 \mathcal{O}_{\mathbb{C}} (holomorphic functions on complex plane)
\mathbb{Q} (rational numbers)𝔽 q(z)\mathbb{F}_q(z) (rational fractions/rational function on affine line 𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q})meromorphic functions on complex plane
pp (prime number/non-archimedean place)x𝔽 px \in \mathbb{F}_p, where zx𝔽 q[z]z - x \in \mathbb{F}_q[z] is the irreducible monic polynomial of degree onexx \in \mathbb{C}, where zx𝒪 z - x \in \mathcal{O}_{\mathbb{C}} is the function which subtracts the complex number xx from the variable zz
\infty (place at infinity)\infty
Spec()Spec(\mathbb{Z}) (Spec(Z))𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q} (affine line)complex plane
Spec()place Spec(\mathbb{Z}) \cup place_{\infty} 𝔽 q\mathbb{P}_{\mathbb{F}_q} (projective line)Riemann sphere
p() p()p\partial_p \coloneqq \frac{(-)^p - (-)}{p} (Fermat quotient)z\frac{\partial}{\partial z} (coordinate derivation)
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
/(p n)\mathbb{Z}/(p^n \mathbb{Z}) (prime power local ring)𝔽 q[z]/((zx) n𝔽 q[z])\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z]) (nn-th order univariate local Artinian 𝔽 q \mathbb{F}_q -algebra)[z]/((zx) n[z])\mathbb{C}[z]/((z-x)^n \mathbb{C}[z]) (nn-th order univariate Weil \mathbb{C} -algebra)
p\mathbb{Z}_p (p-adic integers)𝔽 q[[zx]]\mathbb{F}_q[ [ z -x ] ] (power series around xx)[[zx]]\mathbb{C}[ [z-x] ] (holomorphic functions on formal disk around xx)
Spf( p)×Spec()XSpf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X (“pp-arithmetic jet space” of XX at pp)formal disks in XX
p\mathbb{Q}_p (p-adic numbers)𝔽 q((zx))\mathbb{F}_q((z-x)) (Laurent series around xx)((zx))\mathbb{C}((z-x)) (holomorphic functions on punctured formal disk around xx)
𝔸 = pplace p\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p (ring of adeles)𝔸 𝔽 q((t))\mathbb{A}_{\mathbb{F}_q((t))} ( adeles of function field ) x((zx))\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)
𝕀 =GL 1(𝔸 )\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}}) (group of ideles)𝕀 𝔽 q((t))\mathbb{I}_{\mathbb{F}_q((t))} ( ideles of function field ) xGL 1(((zx)))\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))
theta functions
Jacobi theta function
zeta functions
Riemann zeta functionGoss zeta function
branched covering curves
KK a number field (K\mathbb{Q} \hookrightarrow K a possibly ramified finite dimensional field extension)KK a function field of an algebraic curve Σ\Sigma over 𝔽 p\mathbb{F}_pK ΣK_\Sigma (sheaf of rational functions on complex curve Σ\Sigma)
𝒪 K\mathcal{O}_K (ring of integers)𝒪 Σ\mathcal{O}_{\Sigma} (structure sheaf)
Spec an(𝒪 K)Spec()Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z}) (spectrum with archimedean places)Σ\Sigma (arithmetic curve)ΣP 1\Sigma \to \mathbb{C}P^1 (complex curve being branched cover of Riemann sphere)
() pΦ()p\frac{(-)^p - \Phi(-)}{p} (lift of Frobenius morphism/Lambda-ring structure)z\frac{\partial}{\partial z}
genus of a number fieldgenus of an algebraic curvegenus of a surface
formal neighbourhoods
vv prime ideal in ring of integers 𝒪 K\mathcal{O}_KxΣx \in \SigmaxΣx \in \Sigma
K vK_v (formal completion at vv)((z x))\mathbb{C}((z_x)) (function algebra on punctured formal disk around xx)
𝒪 K v\mathcal{O}_{K_v} (ring of integers of formal completion)[[z x]]\mathbb{C}[ [ z_x ] ] (function algebra on formal disk around xx)
𝔸 K\mathbb{A}_K (ring of adeles) xΣ ((z x))\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x)) (restricted product of function rings on all punctured formal disks around all points in Σ\Sigma)
𝒪\mathcal{O} xΣ[[z x]]\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ] (function ring on all formal disks around all points in Σ\Sigma)
𝕀 K=GL 1(𝔸 K)\mathbb{I}_K = GL_1(\mathbb{A}_K) (group of ideles) xΣ GL 1(((z x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))
Galois theory
Galois groupπ 1(Σ)\pi_1(\Sigma) fundamental group
Galois representationflat connection (“local system”) on Σ\Sigma
class field theory
class field theorygeometric class field theory
Hilbert reciprocity lawArtin reciprocity lawWeil reciprocity law
GL 1(K)\GL 1(𝔸 K)GL_1(K)\backslash GL_1(\mathbb{A}_K) (idele class group)
GL 1(K)\GL 1(𝔸 K)/GL 1(𝒪)GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})Bun GL 1(Σ)Bun_{GL_1}(\Sigma) (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
number field Langlands correspondencefunction field Langlands correspondencegeometric Langlands correspondence
GL n(K)\GL n(𝔸 K)//GL n(𝒪)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()(Σ)Bun_{GL_n(\mathbb{C})}(\Sigma) (moduli stack of bundles on the curve Σ\Sigma, by Weil uniformization theorem)
Tamagawa-Weil for number fieldsTamagawa-Weil for function fields
theta functions
Hecke theta functionfunctional determinant line bundle of Dirac operator/chiral Laplace operator on Σ\Sigma
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface/of the Laplace operator on Σ\Sigma
higher dimensional spaces
zeta functionsHasse-Weil zeta function


Textbook account:

Lecture notes:

See also:

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.

Last revised on December 20, 2021 at 17:15:50. See the history of this page for a list of all contributions to it.