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 + c_n−1xⁿ⁻¹ + … + c₀ . This ring is often denoted by O_K 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 O_K.

The ring Z is the simplest possible ring of integers.^[1] Namely, Z = O_Q 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

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

$x=\sum_{i=1}^na_ib_i,$

with a_i ∈ Z.^[3] The rank n of O_K 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]

Examples

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 O_K = Z[ζ] is given by (1, ζ, ζ², … , ζ^p−2).^[5]

If d is a square-free integer and K = Q(√d) is the corresponding quadratic field, then O_K 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 O_K 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]

Generalization

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 Z_p are the ring of integers of the p-adic numbers Q_p .

References

• Cassels, J.W.S. (1986). Local fields. London Mathematical Society Student Texts 3. Cambridge: Cambridge University Press. ISBN 0-521-31525-5. Zbl 0595.12006.
• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859
• Samuel, Pierre (1972). Algebraic number theory. Hermann/Kershaw.

Notes

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. Cassels (1986) p.192
3. Cassels (1986) p.193
4. Samuel (1972) p.49
5. Samuel (1972) p.43
6. Samuel (1972) p.35
7. Artin, Michael (2011). Algebra. Prentice Hall. p. 360. ISBN 978-0-13-241377-0.
8. Samuel (1972) p.50
9. Samuel (1972) pp.59-62
10. Cassels (1986) p.41