World Library  
Flag as Inappropriate
Email this Article

Ring of integers

Article Id: WHEBN0000460613
Reproduction Date:

Title: Ring of integers  
Author: World Heritage Encyclopedia
Language: English
Subject: Quadratic field, Glossary of arithmetic and Diophantine geometry, Ring (mathematics), Euclidean domain, Integral domain
Collection: Algebraic Number Theory, Ring Theory
Publisher: World Heritage Encyclopedia

Ring of integers

In mathematics, the ring of integers of an algebraic number field K is the ring of all integral elements contained in K. An integral element is a root of a monic polynomial with rational integer coefficients, xn + cn−1xn−1 + … + c0. This ring is often denoted by OK or \mathcal O_K. Since any rational integer number belongs to K and is its integral element, the ring Z is always a subring of OK.

The ring Z is the simplest possible ring of integers.[1] Namely, Z = OQ where Q is the field of rational numbers.[2] And indeed, in algebraic number theory the elements of Z are often called the "rational integers" because of this.

The ring of integers of an algebraic number field is the unique maximal order in the field.


  • Properties 1
  • Examples 2
  • Multiplicative structure 3
  • Generalization 4
  • References 5
  • Notes 6


The ring of integers OK is a finitely-generated Z-module. Indeed it is a free Z-module, and thus has an integral basis, that is a basis b1, … ,bn ∈ OK of the Q-vector space K such that each element x in OK can be uniquely represented as


with aiZ.[3] The rank n of OK as a free Z-module is equal to the degree of K over Q.

The rings of integers in number fields are Dedekind domains.[4]


If p is a prime, ζ is a pth root of unity and K = Q(ζ) is the corresponding cyclotomic field, then an integral basis of OK = Z[ζ] is given by (1, ζ, ζ2, … , ζp−2).[5]

If d is a square-free integer and K = Q(d) is the corresponding quadratic field, then OK is a ring of quadratic integers and its integral basis is given by (1, (1 + d)/2) if d ≡ 1 (mod 4) and by (1, d) if d ≡ 2, 3 (mod 4).[6]

Multiplicative structure

In a ring of integers, every element has a factorisation into irreducible elements, but the ring need not have the property of unique factorisation: for example, in the ring of integers ℤ[√-5] the element 6 has two essentially different factorisations into irreducibles:[7][4]

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

A ring of integers is always a Dedekind domain, and so has unique factorisation of ideals into prime ideals.[8]

The units of a ring of integers OK is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of K. A set of torsion-free generators is called a set of fundamental units.[9]


One defines the ring of integers of a 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.[10] 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.[2]

For example, the p-adic integers Zp are the ring of integers of the p-adic numbers Qp.




  1. ^ The ring of integers, without specifying the field, refers to the ring Z of "ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "integer" in abstract algebra.
  2. ^ a b Cassels (1986) p.192
  3. ^ Cassels (1986) p.193
  4. ^ a b Samuel (1972) p.49
  5. ^ Samuel (1972) p.43
  6. ^ Samuel (1972) p.35
  7. ^ Artin, Michael (2011). Algebra. Prentice Hall. p. 360.  
  8. ^ Samuel (1972) p.50
  9. ^ Samuel (1972) pp.59-62
  10. ^ Cassels (1986) p.41
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.