# Algebraic integer

An **algebraic integer** is a complex number that is a root of some monic polynomial with coefficients in the ordinary integers \( \mathbb{Z} \), that is, a polynomial whose leading coefficient is 1.<sup>[1](https://www.maths.dur.ac.uk/users/jack.g.shotton/Teaching/number-theory/notes.html)</sup> The set of all algebraic integers is denoted \( \overline{\mathbb{Z}} \), and within a number field \( K \) (a finite extension of \( \mathbb{Q} \)) the algebraic integers form the **ring of integers** \( \mathcal{O}_K = \overline{\mathbb{Z}} \cap K \), the analog of \( \mathbb{Z} \) inside \( K \).<sup>[1](https://www.maths.dur.ac.uk/users/jack.g.shotton/Teaching/number-theory/notes.html)</sup><sup> • </sup><sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup>  These rings are the central objects of algebraic number theory, which studies number fields and their rings of integers as generalizations of the rational numbers and \( \mathbb{Z} \).<sup>[3](https://math.ucla.edu/~sharifi/notes/algnum.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \alpha \) is an algebraic integer if \( f(\alpha) = 0 \) for some monic \( f \in \mathbb{Z}[x] \); the set of all of them is \( \overline{\mathbb{Z}} \)<sup>[1](https://www.maths.dur.ac.uk/users/jack.g.shotton/Teaching/number-theory/notes.html)</sup> |
| Ring structure | Sums and products of algebraic integers are algebraic integers, so \( \overline{\mathbb{Z}} \) is a ring<sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup> |
| Minimal polynomial | The minimal polynomial over \( \mathbb{Q} \) of an algebraic integer has coefficients in \( \mathbb{Z} \)<sup>[4](https://crypto.stanford.edu/pbc/notes/numberfield/numberfields.html)</sup> |
| Finite generation criterion | \( \alpha \) is an algebraic integer if and only if \( \mathbb{Z}[\alpha] \) is finitely generated as an abelian group<sup>[5](https://www.ucl.ac.uk/~ucahmki/ant/3704notes.pdf)</sup> |
| Quadratic fields | For square-free \( d \), \( \mathcal{O}_{\mathbb{Q}(\sqrt{d})} = \mathbb{Z}[(1+\sqrt{d})/2] \) if \( d \equiv 1 \bmod 4 \), and \( \mathbb{Z}[\sqrt{d}] \) if \( d \equiv 2, 3 \bmod 4 \)<sup>[6](https://www.math.ucla.edu/~sharifi/notes/algnum-ch01.html)</sup> |
| Rational integers | \( \mathcal{O}_K \cap \mathbb{Q} = \mathbb{Z} \) and \( \mathbb{Q} \cdot \mathcal{O}_K = K \)<sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup> |
| Field of fractions | The quotient field of \( \overline{\mathbb{Z}} \) is \( \overline{\mathbb{Q}} \)<sup>[6](https://math.ucla.edu/~sharifi/notes/algnum.html)</sup> |

## Why the definition takes this form

The definition is designed so that the algebraic integers behave inside \( \overline{\mathbb{Q}} \) the way \( \mathbb{Z} \) behaves inside \( \mathbb{Q} \). Three natural requirements for a notion of "integer" in \( \overline{\mathbb{Q}} \) are that the integers form a ring, that the conjugates of an integer are again integers, and that a rational number is an integer exactly when it is an ordinary integer.<sup>[1](https://www.maths.dur.ac.uk/users/jack.g.shotton/Teaching/number-theory/notes.html)</sup> If \( \alpha \) satisfies these, its conjugates \( \alpha_1, \ldots, \alpha_d \) are integers, so the coefficients of \( p_\alpha(x) = (x - \alpha_1) \cdots (x - \alpha_d) \), being sums and products of conjugates, are integers lying in \( \mathbb{Q} \), hence lie in \( \mathbb{Z} \). So integers should be roots of monic polynomials in \( \mathbb{Z}[x] \), and the definition is chosen to make this true.<sup>[1](https://www.maths.dur.ac.uk/users/jack.g.shotton/Teaching/number-theory/notes.html)</sup>

It is not obvious from the definition that sums and products of algebraic integers are again algebraic integers. For example, \( i \) is a root of \( x^2 + 1 \) and \( (1+\sqrt{-3})/2 \) is a root of \( x^2 - x + 1 \), but nothing about these two polynomials immediately shows that \( i + (1+\sqrt{-3})/2 \) satisfies any monic polynomial over \( \mathbb{Z} \).<sup>[1](https://www.maths.dur.ac.uk/users/jack.g.shotton/Teaching/number-theory/notes.html)</sup>

## Characterizations and basic properties

**Finite generation.** The workable criterion is module-theoretic: \( \alpha \) is an algebraic integer if and only if \( \mathbb{Z}[\alpha] \) is finitely generated as an abelian group.<sup>[5](https://www.ucl.ac.uk/~ucahmki/ant/3704notes.pdf)</sup> If \( \alpha \) satisfies a monic polynomial of degree \( n \), every power \( \alpha^m \) with \( m \ge n \) reduces to a \( \mathbb{Z} \)-combination of \( 1, \alpha, \ldots, \alpha^{n-1} \), so those powers generate \( \mathbb{Z}[\alpha] \).<sup>[5](https://www.ucl.ac.uk/~ucahmki/ant/3704notes.pdf)</sup> Conversely, finite generation of \( \mathbb{Z}[\alpha] \) by \( n \) elements lets one build an \( n \times n \) integer matrix encoding multiplication by \( \alpha \), and the Cayley-Hamilton style determinant \( \det(\alpha I - M) = 0 \) is a monic equation of degree \( n \) for \( \alpha \).<sup>[4](https://crypto.stanford.edu/pbc/notes/numberfield/numberfields.html)</sup>

This criterion proves closure under addition and multiplication. If \( \alpha \) and \( \beta \) are integral, then \( \mathbb{Z}[\alpha] \) and \( \mathbb{Z}[\beta] \) are finitely generated, so \( \mathbb{Z}[\alpha, \beta] \) is finitely generated; \( \mathbb{Z}[\alpha + \beta] \) and \( \mathbb{Z}[\alpha\beta] \) are submodules of it, hence also finitely generated, so \( \alpha + \beta \) and \( \alpha\beta \) are integral.<sup>[4](https://crypto.stanford.edu/pbc/notes/numberfield/numberfields.html)</sup><sup> • </sup><sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup>  Thus \( \overline{\mathbb{Z}} \) is a ring.<sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup>

**Minimal polynomial.** A polynomial with rational coefficients can be scaled to be monic or to have integer coefficients, but the algebraic integer condition requires both at once.<sup>[7](https://notes.edubirdie.com/docs/massachusetts-institute-of-technology/res-18-012-algebra-ii-student-notes/97008-lecture-14-notes-number-fields)</sup> In fact the two conditions are equivalent: if \( \alpha \) is a root of any monic \( f \in \mathbb{Z}[x] \), then \( f \) is a multiple of the minimal polynomial \( m_\alpha \in \mathbb{Q}[x] \), and Gauss's Lemma produces a constant \( c \in \mathbb{Q}^{\times} \) with both \( cq \) and \( c^{-1} m_\alpha \) in \( \mathbb{Z}[x] \); since \( f \) and \( q \) are monic, \( c = \pm 1 \), so \( m_\alpha \in \mathbb{Z}[x] \).<sup>[5](https://www.ucl.ac.uk/~ucahmki/ant/3704notes.pdf)</sup> Equivalently, a monic polynomial in \( \mathbb{Z}[x] \) cannot factor over \( \mathbb{Q} \) into monic factors that fail to have integer coefficients.<sup>[4](https://crypto.stanford.edu/pbc/notes/numberfield/numberfields.html)</sup>

**Rational numbers.** A rational number \( a/b \) in lowest terms has minimal polynomial \( bx - a \), which is monic in \( \mathbb{Z}[x] \) only when \( b = 1 \); so a rational number is an algebraic integer exactly when it is an ordinary integer.<sup>[1](https://www.maths.dur.ac.uk/users/jack.g.shotton/Teaching/number-theory/notes.html)</sup><sup> • </sup><sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup>  The example \( 1/2 \) shows the condition has teeth: \( \mathbb{Z}[1/2] \) is not finitely generated as a \( \mathbb{Z} \)-module, since it is infinite while \( \mathbb{Z}[1/2]/2\,\mathbb{Z}[1/2] = 0 \).<sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup> More generally \( \mathcal{O}_K \cap \mathbb{Q} = \mathbb{Z} \) for every number field, and every element of \( K \) becomes an algebraic integer after multiplication by a suitable ordinary integer, so \( \mathbb{Q} \cdot \mathcal{O}_K = K \).<sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup> At the top level, the quotient field of \( \overline{\mathbb{Z}} \) is \( \overline{\mathbb{Q}} \).<sup>[6](https://math.ucla.edu/~sharifi/notes/algnum.html)</sup>

## Rings of integers of number fields

For a number field \( K \), the ring of integers \( \mathcal{O}_K \) is the subring of \( K \) consisting of all roots of monic polynomials in \( \mathbb{Z}[x] \); equivalently, it is the integral closure of \( \mathbb{Z} \) in \( K \).<sup>[3](https://math.ucla.edu/~sharifi/notes/algnum.html)</sup><sup> • </sup><sup>[6](https://math.ucla.edu/~sharifi/notes/algnum.html)</sup>  The simplest cases are the quadratic fields \( \mathbb{Q}(\sqrt{d}) \) with \( d \) square-free: the ring of integers is \( \mathbb{Z}[(1+\sqrt{d})/2] \) when \( d \equiv 1 \bmod 4 \) and \( \mathbb{Z}[\sqrt{d}] \) when \( d \equiv 2, 3 \bmod 4 \).<sup>[6](https://math.ucla.edu/~sharifi/notes/algnum.html)</sup> Thus \( \mathbb{Q}(i) \) has ring of integers \( \mathbb{Z}[i] \), the Gaussian integers, while \( \mathbb{Q}(\sqrt{5}) \) has \( \mathbb{Z}[(1+\sqrt{5})/2] \), since the golden ratio \( (1+\sqrt{5})/2 \) satisfies \( x^2 - x - 1 \).<sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup>

The naive guess \( \mathcal{O}_K = \mathbb{Z}[\theta] \) for a generator \( \theta \) of \( K \) fails in general: the same field can be written \( \mathbb{Q}(\theta) = \mathbb{Q}(\theta') \) with \( \mathbb{Z}[\theta] \neq \mathbb{Z}[\theta'] \).<sup>[1](https://www.maths.dur.ac.uk/users/jack.g.shotton/Teaching/number-theory/notes.html)</sup> What is always true is that \( \mathbb{Z}[\alpha] \) is an **order** in \( \mathcal{O}_K \) when \( \alpha \) is an algebraic integer, meaning a subring of finite index as an abelian group; but frequently \( \mathbb{Z}[\alpha] \neq \mathcal{O}_K \).<sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup> Orders are used in practice because they are easier to write down explicitly than \( \mathcal{O}_K \) itself.<sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup> Determining the full ring of integers, for instance the index of \( \mathbb{Z}[\theta] \) inside \( \mathcal{O}_K \) for fields like \( \mathbb{Q}(\sqrt[12]{a}) \), remains an active computational problem.<sup>[8](https://arxiv.org/abs/2304.00339)</sup>

## Why rings of integers matter: factorization

Unlike \( \mathbb{Z} \), not all rings of integers are unique factorization domains. The standard example is \( \mathbb{Z}[\sqrt{-5}] \), the ring of integers of \( \mathbb{Q}(\sqrt{-5}) \), in which

\[ 6 = 2 \cdot 3 = (1 - \sqrt{-5})(1 + \sqrt{-5}) \]

gives two distinct factorizations of 6 into irreducibles.<sup>[3](https://math.ucla.edu/~sharifi/notes/algnum.html)</sup><sup> • </sup><sup>[9](https://arxiv.org/abs/1711.10842)</sup>  The failure of unique factorization of elements motivated the development of ideal theory: [Richard Dedekind](https://www.edgepedia.org/richard-dedekind), known for his work on algebraic numbers, fields, and modules, replaced Ernst Kummer's "ideal numbers" with set-theoretic **ideals**, subsets of the number field or of its ring of integers, which restore a form of unique factorization at the level of ideals.<sup>[10](https://plato.stanford.edu/entries/dedekind-foundations/)</sup> In the specific case of \( \mathbb{Z}[\sqrt{-5}] \), the ring is not a UFD but does satisfy the weaker property of half-factoriality, meaning any two factorizations of an element into irreducibles have the same length.<sup>[9](https://arxiv.org/abs/1711.10842)</sup>

The ring of integers also supports the local machinery of the subject: for a prime ideal \( \mathfrak{p} \) of \( \mathcal{O} \), one defines a metric on \( F \) measuring the highest power of \( \mathfrak{p} \) dividing the difference of two points, and the completion of \( F \) with respect to this metric is a \( p \)-adic field.<sup>[3](https://math.ucla.edu/~sharifi/notes/algnum.html)</sup>

## Examples and non-examples

- \( \sqrt{2} \) is an algebraic integer, being a root of \( x^2 - 2 \).<sup>[5](https://www.ucl.ac.uk/~ucahmki/ant/3704notes.pdf)</sup>
- \( i \) and \( (1+\sqrt{-3})/2 \) are algebraic integers, with monic polynomials \( x^2 + 1 \) and \( x^2 - x + 1 \).<sup>[1](https://www.maths.dur.ac.uk/users/jack.g.shotton/Teaching/number-theory/notes.html)</sup>
- \( 1/2 \) is not an algebraic integer.<sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup>
- Every ordinary integer is an algebraic integer, and \( \mathcal{O}_K \cap \mathbb{Q} = \mathbb{Z} \) shows these are the only rational ones.<sup>[2](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)</sup>

## References

1. [Number Theory III: Notes, Jack Shotton, Durham University](https://www.maths.dur.ac.uk/users/jack.g.shotton/Teaching/number-theory/notes.html)
2. [Rings of Algebraic Integers, Math 129 notes, Harvard University](https://abel.math.harvard.edu/archive/129_spring_04/ant/html/node14.html)
3. [Algebraic Number Theory — Introduction, Romyar Sharifi, UCLA](https://math.ucla.edu/~sharifi/notes/algnum.html)
4. [Introduction — Number Fields, Stanford University notes](https://crypto.stanford.edu/pbc/notes/numberfield/numberfields.html)
5. [Math 3704 Lecture Notes, University College London](https://www.ucl.ac.uk/~ucahmki/ant/3704notes.pdf)
6. [Algebraic Number Theory — Chapter 1 Abstract algebra, Romyar Sharifi, UCLA](https://www.math.ucla.edu/~sharifi/notes/algnum-ch01.html)
7. [Lecture 14 Notes: Number Fields, MIT 18.012](https://notes.edubirdie.com/docs/massachusetts-institute-of-technology/res-18-012-algebra-ii-student-notes/97008-lecture-14-notes-number-fields)
8. [Discriminant and integral basis of \( \mathbb{Q}(\sqrt[12]{a}) \), arXiv:2304.00339](https://arxiv.org/abs/2304.00339)
9. [How do elements really factor in \( \mathbb{Z}[\sqrt{-5}] \)?, arXiv:1711.10842](https://arxiv.org/abs/1711.10842)
10. [Dedekind's Contributions to the Foundations of Mathematics, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/dedekind-foundations/)

---
*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*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license