# Order (ring theory)

In ring theory and algebraic number theory, an **order** is a subring of the ring of integers of a number field that has finite index in that ring, equivalently a subring that is a free abelian group of rank equal to the field degree. Orders are the rings in which the arithmetic of algebraic integers is actually computed: the full ring of integers is often hard to write down, while an order such as \( \mathbf{Z}[\alpha] \) is immediate from a defining polynomial.<sup>[1](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | An order in \( \mathcal{O}_K \) is a subring \( R \) of the ring of integers such that the quotient \( \mathcal{O}_K/R \) of abelian groups is finite; \( R \) must contain 1.<sup>[1](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup> |
| Equivalent form | An order of \( K \) is a subring of \( \mathcal{O}_K \) that is free of rank \( [K:\mathbf{Q}] \) as a \( \mathbf{Z} \)-module.<sup>[2](https://docs.hecke.thofma.com/v0.17/orders/introduction/)</sup> |
| Maximal order | The ring of integers \( \mathcal{O}_K \), the set of all algebraic integers in \( K \), is itself an order, called the maximal order; it is a Dedekind domain with unique factorization of nonzero integral ideals.<sup>[3](https://gskopp.com/wp-content/uploads/2022/12/cftfororders_kopplagarias.pdf)</sup> |
| Non-maximal orders | Orders other than \( \mathcal{O}_K \) are never Dedekind domains: not all integral ideals factor uniquely into prime ideals, and not all fractional ideals are invertible.<sup>[3](https://gskopp.com/wp-content/uploads/2022/12/cftfororders_kopplagarias.pdf)</sup> |
| Conductor | Each order \( \mathcal{O} \) has a conductor, the ideal \( \mathfrak{f} = \{a \in \mathcal{O}_K : a\mathcal{O}_K \subseteq \mathcal{O}\} \), which is nonzero and is the greatest ideal of \( \mathcal{O}_K \) lying in \( \mathcal{O} \).<sup>[4](https://www.researchgate.net/publication/334770842_Maximality_of_orders_in_number_fields)</sup> |
| Quadratic case | In an imaginary quadratic field the orders are exactly the rings \( \mathbf{Z} + f\mathcal{O}_K \) for positive integers \( f \), and the conductor \( f \) equals the index \( [\mathcal{O}_K : \mathcal{O}] \).<sup>[5](https://math.mit.edu/classes/18.783/2025/LectureNotes12.pdf)</sup> |
| Discriminant relation | For an order \( \mathcal{O} \) with integral closure \( B \), the discriminants satisfy \( D_{\mathcal{O}} = [B:\mathcal{O}]^{2} D_B \).<sup>[6](https://math.mit.edu/classes/18.785/2018fa/LectureNotes12.pdf)</sup> |

## Definition and first examples

Let \( K \) be a number field, a finite extension of \( \mathbf{Q} \). Its ring of integers is

\[ \mathcal{O}_K = K \cap \overline{\mathbf{Z}} = \{x \in K : x \text{ is an algebraic integer}\}. \]

An order in \( \mathcal{O}_K \) is any subring \( R \) of \( \mathcal{O}_K \) for which the quotient \( \mathcal{O}_K/R \) of abelian groups is finite.<sup>[1](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup> Because a ring in this convention contains 1, every order contains \( \mathbf{Z} \), and the finite-index condition forces \( R \) to be a lattice of full rank \( [K:\mathbf{Q}] \) in \( K \).<sup>[2](https://docs.hecke.thofma.com/v0.17/orders/introduction/)</sup>

The basic examples come from quadratic fields. The field \( \mathbf{Q}(i) \) has ring of integers \( \mathbf{Z}[i] \), the Gaussian integers. For every nonzero integer \( n \), the subring \( \mathbf{Z} + ni\mathbf{Z} \) of \( \mathbf{Z}[i] \) is an order. Two nearby subrings fail the definition in instructive ways: \( \mathbf{Z} \) itself is not an order in \( \mathbf{Z}[i] \) because it does not have finite index there, and the subgroup \( 2\mathbf{Z} + i\mathbf{Z} \) is not an order because it is not closed under multiplication.<sup>[1](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup>

**Orders from generators.** If \( K = \mathbf{Q}(\alpha) \) with \( \alpha \) an algebraic integer, then \( \mathbf{Z}[\alpha] \) is an order in \( \mathcal{O}_K \), but frequently \( \mathbf{Z}[\alpha] \neq \mathcal{O}_K \).<sup>[1](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup> This gap between the equation order and the maximal order is the practical reason the concept exists: the equation order is easy to write down, while the maximal order requires real computation.<sup>[7](https://docs.magma-maths.org/GlobalFields/NumberFieldsAndOrders/creation.html)</sup> In the example documented in the Magma handbook, the equation order \( \mathbf{Z}[\alpha] \) is a subring of index \( 2^{9} \cdot 5^{3} \) in the maximal order.<sup>[7](https://docs.magma-maths.org/GlobalFields/NumberFieldsAndOrders/creation.html)</sup>

## The maximal order and the conductor

The ring of integers \( \mathcal{O}_K \) contains every order and is the unique maximal one. As a ring it is a [Dedekind domain](https://www.edgepedia.org/dedekind-domain): every nonzero integral ideal factors uniquely into prime ideals, all nonzero fractional ideals are invertible, and they form a free abelian group with an [ideal class group](https://www.edgepedia.org/ideal-class-group) defined as the quotient by principal ideals.<sup>[3](https://gskopp.com/wp-content/uploads/2022/12/cftfororders_kopplagarias.pdf)</sup>

A non-maximal order loses these properties. Its ideal theory differs from the maximal order in three ways recorded in the class-field-theory literature: not all integral ideals factor uniquely into prime ideals; not all fractional ideals are invertible, so the fractional ideals form only a monoid under multiplication; and the group of invertible fractional ideals need not be free, so in place of an ideal class group one works with an ideal class monoid.<sup>[3](https://gskopp.com/wp-content/uploads/2022/12/cftfororders_kopplagarias.pdf)</sup> In a non-maximal order the ideals dividing the conductor are not invertible, and factorization fails exactly there.<sup>[8](https://www.kevos.com/engineering/orders-and-ideals-in-number-fields)</sup>

The conductor measures how far an order sits from maximality. For an order \( \mathcal{O} \subset \mathcal{O}_K \) it is the ideal

\[ \mathfrak{f} = \{a \in \mathcal{O}_K : a\mathcal{O}_K \subseteq \mathcal{O}\}, \]

which is nonzero and is the greatest ideal of \( \mathcal{O}_K \) contained in \( \mathcal{O} \).<sup>[4](https://www.researchgate.net/publication/334770842_Maximality_of_orders_in_number_fields)</sup> In the quadratic case \( K = \mathbf{Q}(\sqrt{d}) \) with \( d \) square-free, the maximal order is \( \mathbf{Z}[\sqrt{d}] \) when \( d \not\equiv 1 \pmod 4 \) and \( \mathbf{Z}[(1+\sqrt{d})/2] \) when \( d \equiv 1 \pmod 4 \); the order of conductor \( f \) is \( \mathbf{Z}[f\sqrt{d}] \) or \( \mathbf{Z}[f(1+\sqrt{d})/2] \) accordingly, and its conductor is the principal ideal generated by \( f \).<sup>[4](https://www.researchgate.net/publication/334770842_Maximality_of_orders_in_number_fields)</sup> In an imaginary quadratic field the orders are precisely \( \mathbf{Z} + f\mathcal{O}_K \), and the conductor \( f \) equals the index \( [\mathcal{O}_K:\mathcal{O}] \); the same classification holds for real quadratic fields.<sup>[5](https://math.mit.edu/classes/18.783/2025/LectureNotes12.pdf)</sup>

## Discriminants and maximality

The discriminant of an order records its index. If \( \mathcal{O} \) has integral closure \( B \), then

\[ D_{\mathcal{O}} = [B:\mathcal{O}]^{2} D_B, \]

so the order discriminant is the field discriminant multiplied by a perfect square.<sup>[6](https://math.mit.edu/classes/18.785/2018fa/LectureNotes12.pdf)</sup> For an order of the form \( \mathbf{Z}[\alpha] \) the discriminant is generated by \( \mathrm{disc}(f) \) for the minimal polynomial \( f \) of \( \alpha \), and a squarefree \( \mathrm{disc}(f) \) forces \( \mathbf{Z}[\alpha] \) to be the full ring of integers.<sup>[6](https://math.mit.edu/classes/18.785/2018fa/LectureNotes12.pdf)</sup> An equivalent characterization runs through integral closedness: an order is integrally closed if and only if it equals the maximal order.<sup>[9](https://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/order.html)</sup>

## Class groups and ring class fields

Orders carry their own class groups. The [Picard group](https://www.edgepedia.org/picard-group) \( \mathrm{Pic}(\mathcal{O}) \), generated by invertible ideals modulo principal ideals, is finite, and for the maximal order it coincides with the ideal class group of \( K \).<sup>[4](https://www.researchgate.net/publication/334770842_Maximality_of_orders_in_number_fields)</sup> For an order \( \mathfrak{o} \) in an arbitrary number field one can construct a ring class field whose [Galois group](https://www.edgepedia.org/galois-group) is isomorphic to the Picard group of the order, generalizing the classical correspondence known for orders in imaginary quadratic fields and CM-fields.<sup>[10](https://arxiv.org/abs/1405.5776)</sup> These class fields have arithmetic uses: for suitable imaginary quadratic fields, the solvability of \( p = x^{2} + ny^{2} \) for a prime \( p \) reduces to whether \( -n \) is a square modulo \( p \).<sup>[10](https://arxiv.org/abs/1405.5776)</sup>

## History

The concept is due to [Richard Dedekind](https://www.edgepedia.org/richard-dedekind). Having defined the domain of algebraic integers of \( \mathbf{Q}(\alpha) \), in which he proved his theorem on unique decomposition of ideals, he considered more generally the sets of integers of \( \mathbf{Q}(\alpha) \) closed under addition, subtraction, and multiplication, and called them "orders", the domain of integers being the largest order. This was an essentially axiomatic definition of a commutative ring, albeit in a concrete setting.<sup>[11](https://www.researchgate.net/publication/225767243_From_Numbers_to_Rings_The_Early_History_of_Ring_Theory)</sup>

## Computation and counting

Orders are standard objects in computer algebra. Magma computes maximal orders with the Round 2 or Round 4 algorithm, while the equation order requires no computation at all; PARI/GP can compute an order that is maximal at a chosen set of primes without being globally maximal.<sup>[7](https://docs.magma-maths.org/GlobalFields/NumberFieldsAndOrders/creation.html)</sup><sup> • </sup><sup>[12](https://pari.math.u-bordeaux.fr/dochtml/html-stable/usersch6.html)</sup> SageMath represents orders with their conductors and provides tests such as integrally closedness and suborder relations.<sup>[9](https://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/order.html)</sup> The Hecke system treats orders in relative extensions \( L/K \) as finitely generated [torsion-free modules](https://www.edgepedia.org/torsion-free-module) over the Dedekind domain \( \mathcal{O}_K \).<sup>[2](https://docs.hecke.thofma.com/v0.17/orders/introduction/)</sup>

Orders are also counted. Kaplan, Marcinek, and Takloo-Bighash give an asymptotic formula for the number of orders of bounded discriminant contained in the ring of integers of a quintic number field.<sup>[13](https://arxiv.org/abs/1408.1374)</sup> More recently, a fundamental theorem of orders expresses any order in a number field uniquely as an intersection of irreducible orders, along which the index in the ring of integers distributes multiplicatively and the conductor factors into pairwise coprime ideals.<sup>[14](https://arxiv.org/abs/2411.10917)</sup>

## References

1. [Rings of Algebraic Integers, Harvard Math 129 lecture notes (Definition 5.1.6, 5.1.7)](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)
2. [Introduction to orders, Hecke documentation v0.17](https://docs.hecke.thofma.com/v0.17/orders/introduction/)
3. [G. Koppla, Class Field Theory for Orders](https://gskopp.com/wp-content/uploads/2022/12/cftfororders_kopplagarias.pdf)
4. [B. Rothkegel, Maximality of orders in number fields](https://www.researchgate.net/publication/334770842_Maximality_of_orders_in_number_fields)
5. [MIT 18.783 Elliptic Curves, Lecture 12 notes (Theorem 12.27)](https://math.mit.edu/classes/18.783/2025/LectureNotes12.pdf)
6. [MIT 18.785 Number Theory I, Lecture 12 notes](https://math.mit.edu/classes/18.785/2018fa/LectureNotes12.pdf)
7. [Creation Functions, Number Fields and Orders, Magma Handbook](https://docs.magma-maths.org/GlobalFields/NumberFieldsAndOrders/creation.html)
8. [Orders and Ideals in Number Fields, KEVOS](https://www.kevos.com/engineering/orders-and-ideals-in-number-fields)
9. [Orders in number fields, SageMath reference manual](https://doc.sagemath.org/html/en/reference/number_fields/sage/rings/number_field/order.html)
10. [On Orders in Number Fields: Picard Groups, Ring Class Fields and Applications, arXiv:1405.5776](https://arxiv.org/abs/1405.5776)
11. [From Numbers to Rings: The Early History of Ring Theory, Elem. Math. 53 (1998)](https://www.researchgate.net/publication/225767243_From_Numbers_to_Rings_The_Early_History_of_Ring_Theory)
12. [PARI/GP Manual, usersch6: Maximal order and discriminant](https://pari.math.u-bordeaux.fr/dochtml/html-stable/usersch6.html)
13. [N. Kaplan, J. Marcinek, R. Takloo-Bighash, Distribution of orders in number fields, arXiv:1408.1374](https://arxiv.org/abs/1408.1374)
14. [G. D. Patil, Weighted enumeration of Number Fields using Pseudo and Sudo Maximal Orders, arXiv:2411.10917](https://arxiv.org/abs/2411.10917)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Ring theory › Factorization and orders › Orders in rings and rings of integers*

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