rational integers in ideals

Any non-zero ideal of an algebraic number fieldMathworldPlanetmath K, i.e. of the maximal orderPlanetmathPlanetmath 𝒪K of K, contains positive rational integers.

Proof.  Let  𝔞(0)  be any ideal of 𝒪K.  Take a nonzero element α of 𝔞.  The norm (http://planetmath.org/NormInNumberField) of α is the product


where n is the degree of the number field and α(1),α(2),,α(n) is the set of the http://planetmath.org/node/12046K-conjugatesPlanetmathPlanetmath of  α=α(1).  The number


belongs to the field K and it is an algebraic integerMathworldPlanetmath, since α(2),,α(n) are, as algebraic conjugates of α, also algebraic integers.  Thus  γ𝒪K.  Consequently, the non-zero integer


belongs to the ideal 𝔞, and similarly its opposite number.  So, 𝔞 contains positive integers, in fact infinitely many.

Title rational integers in ideals
Canonical name RationalIntegersInIdeals
Date of creation 2013-03-22 19:08:47
Last modified on 2013-03-22 19:08:47
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 12F05
Classification msc 06B10
Classification msc 11R04
Related topic CharacteristicPolynomialOfAlgebraicNumber
Related topic IdealNorm