examples of ring of integers of a number field

Definition 1.

Let K be a number fieldMathworldPlanetmath. The ring of integersMathworldPlanetmath of K, usually denoted by OK, is the set of all elements αK which are roots of some monic polynomial with coefficients in Z, i.e. those αK which are integral over Z. In other words, OK is the integral closure of Z in K.

Example 1.

Notice that the only rational numbers which are roots of monic polynomials with integer coefficients are the integers themselves. Thus, the ring of integers of is .

Example 2.

Let 𝒪K denote the ring of integers of K=(d), where d is a square-free integer. Then:

𝒪K{1+d2, if d1mod 4,d, if d2,3mod 4.

In other words, if we let

α={1+d2, if d1mod 4,d, if d2,3mod 4.


Example 3.

Let K=(ζn) be a cyclotomic extension of , where ζn is a primitive nth root of unityMathworldPlanetmath. Then the ring of integers of K is 𝒪K=[ζn], i.e.

Example 4.

Let α be an algebraic integerMathworldPlanetmath and let K=(α). It is not true in general that 𝒪K=[α] (as we saw in Example 2, for d1mod4).

Example 5.

Let p be a prime numberMathworldPlanetmath and let F=(ζp) be a cyclotomic extension of , where ζp is a primitive pth root of unity. Let F+ be the maximal real subfieldMathworldPlanetmath of F. It can be shown that:


Moreover, it can also be shown that the ring of integers of F+ is 𝒪F+=[ζp+ζp-1].

Title examples of ring of integers of a number field
Canonical name ExamplesOfRingOfIntegersOfANumberField
Date of creation 2013-03-22 15:08:09
Last modified on 2013-03-22 15:08:09
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Example
Classification msc 13B22
Related topic NumberField
Related topic AlgebraicNumberTheory
Related topic CanonicalBasis
Related topic IntegralBasisOfQuadraticField