# Ring of integers

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Algebraic construction

Algebraic structure → Ring theory Ring theory Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring Z {\displaystyle \mathbb {Z} } • Terminal ring 0 = Z / 1 Z {\displaystyle 0=\mathbb {Z} /1\mathbb {Z} } Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Integers modulo n • Ring of integers • p-adic integers Z p {\displaystyle \mathbb {Z} _{p}} • p-adic numbers Q p {\displaystyle \mathbb {Q} _{p}} • Prüfer p-ring Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty })} Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra v t e

In [mathematics](/source/Mathematics), the **ring of integers** of an [algebraic number field](/source/Algebraic_number_field) K {\displaystyle K} (also sometimes called the **number ring** corresponding to number field *K {\displaystyle K}*)[1] is the [ring](/source/Ring_(mathematics)) of all [algebraic integers](/source/Algebraic_integer) contained in K {\displaystyle K} .[2] An algebraic integer is a [root](/source/Root_of_a_polynomial) of a [monic polynomial](/source/Monic_polynomial) with [integer](/source/Integer) [coefficients](/source/Coefficient): x n + c n − 1 x n − 1 + ⋯ + c 0 {\displaystyle x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}} .[3] This ring is often denoted by O K {\displaystyle O_{K}} or O K {\displaystyle {\mathcal {O}}_{K}} . Since any [integer](/source/Integer) belongs to K {\displaystyle K} and is an [integral element](/source/Integral_element) of K {\displaystyle K} , the ring Z {\displaystyle \mathbb {Z} } is always a [subring](/source/Subring) of O K {\displaystyle O_{K}} .

The ring of integers Z {\displaystyle \mathbb {Z} } is the simplest possible ring of integers.[a] Namely, Z = O Q {\displaystyle \mathbb {Z} =O_{\mathbb {Q} }} where Q {\displaystyle \mathbb {Q} } is the [field](/source/Field_(mathematics)) of [rational numbers](/source/Rational_number).[4] And indeed, in [algebraic number theory](/source/Algebraic_number_theory) the elements of Z {\displaystyle \mathbb {Z} } are often called the "rational integers" because of this.

The next simplest example is the ring of [Gaussian integers](/source/Gaussian_integer) Z [ i ] {\displaystyle \mathbb {Z} [i]} , consisting of [complex numbers](/source/Complex_number) whose [real and imaginary parts](/source/Real_and_imaginary_parts) are integers. It is the ring of integers in the number field Q ( i ) {\displaystyle \mathbb {Q} (i)} of [Gaussian rationals](/source/Gaussian_rational), consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, Z [ i ] {\displaystyle \mathbb {Z} [i]} is a [Euclidean domain](/source/Euclidean_domain).

The ring of integers of an algebraic number field is the unique maximal [order](/source/Order_(ring_theory)) in the field. It is always a [Dedekind domain](/source/Dedekind_domain).[5]

## Properties

The ring of integers O*K* is a [finitely-generated](/source/Finitely-generated_module) Z {\displaystyle \mathbb {Z} } -[module](/source/Module_(mathematics)). Indeed, it is a [free](/source/Free_module) Z {\displaystyle \mathbb {Z} } -module, and thus has an **integral basis**, that is a [basis](/source/Basis_(linear_algebra)) *b*1, ..., *b**n* ∈ O*K* of the Q {\displaystyle \mathbb {Q} } -[vector space](/source/Vector_space) K such that each element x in O*K* can be uniquely represented as

- x = ∑ i = 1 n a i b i , {\displaystyle x=\sum _{i=1}^{n}a_{i}b_{i},}

with a i ∈ Z {\displaystyle a_{i}\in \mathbb {Z} } .[6] The [rank](/source/Rank_of_a_free_module) n of O*K* as a free Z {\displaystyle \mathbb {Z} } -module is equal to the [degree](/source/Degree_of_a_field_extension) of K over Q {\displaystyle \mathbb {Q} } .

## Examples

### Computational tool

A useful tool for computing the integral closure of the ring of integers in an algebraic field K / Q {\displaystyle K/\mathbb {Q} } is the [discriminant](/source/Discriminant_of_an_algebraic_number_field). If *K* is of degree *n* over Q {\displaystyle \mathbb {Q} } , and α 1 , … , α n ∈ O K {\displaystyle \alpha _{1},\ldots ,\alpha _{n}\in {\mathcal {O}}_{K}} form a basis of K {\displaystyle K} over Q {\displaystyle \mathbb {Q} } , set d = Δ K / Q ( α 1 , … , α n ) {\displaystyle d=\Delta _{K/\mathbb {Q} }(\alpha _{1},\ldots ,\alpha _{n})} . Then, O K {\displaystyle {\mathcal {O}}_{K}} is a [submodule](/source/Submodule) of the Z {\displaystyle \mathbb {Z} } -module spanned by α 1 / d , … , α n / d {\displaystyle \alpha _{1}/d,\ldots ,\alpha _{n}/d} .[7] pg. 33 In fact, if *d* is square-free, then α 1 , … , α n {\displaystyle \alpha _{1},\ldots ,\alpha _{n}} forms an integral basis for O K {\displaystyle {\mathcal {O}}_{K}} .[7] pg. 35

### Cyclotomic extensions

If p is a [prime](/source/Prime_number), *ζ* is a pth [root of unity](/source/Root_of_unity) and K = Q ( ζ ) {\displaystyle K=\mathbb {Q} (\zeta )} is the corresponding [cyclotomic field](/source/Cyclotomic_field), then an integral basis of O K = Z [ ζ ] {\displaystyle {\mathcal {O}}_{K}=\mathbb {Z} [\zeta ]} is given by (1, *ζ*, *ζ* 2, ..., *ζ* *p*−2).[8]

### Quadratic extensions

If d {\displaystyle d} is a [square-free integer](/source/Square-free_integer) and K = Q ( d ) {\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)} is the corresponding [quadratic field](/source/Quadratic_field), then O K {\displaystyle {\mathcal {O}}_{K}} is a ring of [quadratic integers](/source/Quadratic_integer) and its integral basis is given by ( 1 , 1 + d 2 ) {\displaystyle \left(1,{\frac {1+{\sqrt {d}}}{2}}\right)} if *d* ≡ 1 ([mod](/source/Modular_arithmetic) 4) and by ( 1 , d ) {\displaystyle (1,{\sqrt {d}})} if *d* ≡ 2, 3 (mod 4).[9] This can be found by computing the [minimal polynomial](/source/Minimal_polynomial_(field_theory)) of an arbitrary element a + b d ∈ Q ( d ) {\displaystyle a+b{\sqrt {d}}\in \mathbb {Q} ({\sqrt {d}})} where a , b ∈ Q {\displaystyle a,b\in \mathbb {Q} } .

## Multiplicative structure

In a ring of integers, every element has a factorization into [irreducible elements](/source/Irreducible_element), but the ring need not have the property of [unique factorization](/source/Unique_factorization_domain): for example, in the ring of integers Z [ − 5 ] {\displaystyle \mathbb {Z} [{\sqrt {-5}}]} , the element 6 has two essentially different factorizations into irreducibles:[5][10]

- 6 = 2 ⋅ 3 = ( 1 + − 5 ) ( 1 − − 5 ) . {\displaystyle 6=2\cdot 3=(1+{\sqrt {-5}})(1-{\sqrt {-5}}).}

A ring of integers is always a [Dedekind domain](/source/Dedekind_domain), and so has unique factorization of [ideals](/source/Ideal_(ring_theory)) into [prime ideals](/source/Prime_ideal).[11]

The [units](/source/Unit_(ring_theory)) of a ring of integers *O**K* is a [finitely generated abelian group](/source/Finitely_generated_abelian_group) by [Dirichlet's unit theorem](/source/Dirichlet's_unit_theorem). The [torsion subgroup](/source/Torsion_subgroup) consists of the [roots of unity](/source/Roots_of_unity) of *K*. A set of torsion-free generators is called a set of *[fundamental units](/source/Fundamental_unit_(number_theory))*.[12]

## Generalization

One defines the ring of integers of a [non-archimedean local field](/source/Non-archimedean_local_field) *F* as the set of all elements of *F* with absolute value ≤ 1; this is a ring because of the strong triangle inequality.[13] If *F* is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.[4]

For example, the [p-adic integers](/source/P-adic_integer) Z p {\displaystyle \mathbb {Z} _{p}} are the ring of integers of the [p-adic numbers](/source/P-adic_number) Q p {\displaystyle \mathbb {Q} _{p}} .

## See also

- [Minimal polynomial (field theory)](/source/Minimal_polynomial_(field_theory))

- [Integral closure](/source/Integral_closure) – gives a technique for computing integral closures

## Notes

1. **[^](#cite_ref-4)** *The ring of integers*, without specifying the field, refers to the ring Z {\displaystyle \mathbb {Z} } of "ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "[integer](https://en.wiktionary.org/wiki/integer)" in abstract algebra.

## Citations

1. **[^](#cite_ref-1)** Marcus, Daniel A. (2018). *Number fields*. Universitext. Emanuele Sacco (2nd ed.). Cham: Springer. [ISBN](/source/ISBN_(identifier)) [978-3-319-90232-6](https://en.wikipedia.org/wiki/Special:BookSources/978-3-319-90232-6).

1. **[^](#cite_ref-FOOTNOTEAlacaWilliams2003110Defs._6.1.2-3_2-0)** [Alaca & Williams 2003](#CITEREFAlacaWilliams2003), p. 110, Defs. 6.1.2-3.

1. **[^](#cite_ref-FOOTNOTEAlacaWilliams200374Defs._4.1.1-2_3-0)** [Alaca & Williams 2003](#CITEREFAlacaWilliams2003), p. 74, Defs. 4.1.1-2.

1. ^ [***a***](#cite_ref-FOOTNOTECassels1986192_5-0) [***b***](#cite_ref-FOOTNOTECassels1986192_5-1) [Cassels 1986](#CITEREFCassels1986), p. 192.

1. ^ [***a***](#cite_ref-FOOTNOTESamuel197249_6-0) [***b***](#cite_ref-FOOTNOTESamuel197249_6-1) [Samuel 1972](#CITEREFSamuel1972), p. 49.

1. **[^](#cite_ref-Cas193_7-0)** Cassels (1986) p. 193

1. ^ [***a***](#cite_ref-:0_8-0) [***b***](#cite_ref-:0_8-1) Baker. ["Algebraic Number Theory"](http://people.math.gatech.edu/~mbaker/pdf/ANTBook.pdf) (PDF). pp. 33–35.

1. **[^](#cite_ref-FOOTNOTESamuel197243_9-0)** [Samuel 1972](#CITEREFSamuel1972), p. 43.

1. **[^](#cite_ref-FOOTNOTESamuel197235_10-0)** [Samuel 1972](#CITEREFSamuel1972), p. 35.

1. **[^](#cite_ref-11)** Artin, Michael (2011). *Algebra*. Prentice Hall. p. 360. [ISBN](/source/ISBN_(identifier)) [978-0-13-241377-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-13-241377-0).

1. **[^](#cite_ref-FOOTNOTESamuel197250_12-0)** [Samuel 1972](#CITEREFSamuel1972), p. 50.

1. **[^](#cite_ref-FOOTNOTESamuel197259–62_13-0)** [Samuel 1972](#CITEREFSamuel1972), pp. 59–62.

1. **[^](#cite_ref-FOOTNOTECassels198641_14-0)** [Cassels 1986](#CITEREFCassels1986), p. 41.

## References

- Alaca, Saban; Williams, Kenneth S. (2003). [*Introductory Algebraic Number Theory*](https://www.cambridge.org/core/books/introductory-algebraic-number-theory/9F53B233CD4D717B1A31ECD117FFEA7D). [Cambridge University Press](/source/Cambridge_University_Press). [ISBN](/source/ISBN_(identifier)) [9780511791260](https://en.wikipedia.org/wiki/Special:BookSources/9780511791260).

- [Cassels, J.W.S.](/source/J._W._S._Cassels) (1986). [*Local fields*](https://books.google.com/books?id=UY52SQnV9w4C&q=%22ring+of+integers%22). London Mathematical Society Student Texts. Vol. 3. Cambridge: [Cambridge University Press](/source/Cambridge_University_Press). [ISBN](/source/ISBN_(identifier)) [0-521-31525-5](https://en.wikipedia.org/wiki/Special:BookSources/0-521-31525-5). [Zbl](/source/Zbl_(identifier)) [0595.12006](https://zbmath.org/?format=complete&q=an:0595.12006).

- [Neukirch, Jürgen](/source/J%C3%BCrgen_Neukirch) (1999). *Algebraische Zahlentheorie*. *Grundlehren der mathematischen Wissenschaften*. Vol. 322. Berlin: [Springer-Verlag](/source/Springer_Science%2BBusiness_Media). [ISBN](/source/ISBN_(identifier)) [978-3-540-65399-8](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-65399-8). [MR](/source/MR_(identifier)) [1697859](https://mathscinet.ams.org/mathscinet-getitem?mr=1697859). [Zbl](/source/Zbl_(identifier)) [0956.11021](https://zbmath.org/?format=complete&q=an:0956.11021).

- [Samuel, Pierre](/source/Pierre_Samuel) (1972). *Algebraic number theory*. Hermann/Kershaw.

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