K
K
n+c | |
x | |
n-1 |
xn-1+ … +c0
OK
lOK
K
K
Z
OK
The ring of integers
Z
Z=OQ
Q
Z
The next simplest example is the ring of Gaussian integers
Z[i]
Q(i)
Z[i]
The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.
The ring of integers is a finitely-generated -module. Indeed, it is a free -module, and thus has an integral basis, that is a basis of the -vector space such that each element in can be uniquely represented as
na | |
x=\sum | |
ib |
i,
A useful tool for computing the integral closure of the ring of integers in an algebraic field is the discriminant. If is of degree over, and
\alpha1,\ldots,\alphan\inl{O}K
d=\DeltaK/Q(\alpha1,\ldots,\alphan)
l{O}K
\alpha1/d,\ldots,\alphan/d
\alpha1,\ldots,\alphan
l{O}K
If is a prime, is a th root of unity and is the corresponding cyclotomic field, then an integral basis of is given by .
If
d
K=Q(\sqrt{d})
l{O}K
a+b\sqrt{d}\inQ(\sqrt{d})
a,b\inQ
In a ring of integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorization: for example, in the ring of integers, the element 6 has two essentially different factorizations into irreducibles:[3]
6=2 ⋅ 3=(1+\sqrt{-5})(1-\sqrt{-5}).
A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals.
The units of a ring of integers is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of . A set of torsion-free generators is called a set of fundamental units.
One defines the ring of integers of a non-archimedean local field as the set of all elements of with absolute value ; this is a ring because of the strong triangle inequality. If 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.
For example, the -adic integers are the ring of integers of the -adic numbers .