rational integers in ideals

Any non-zero ideal of an algebraic number field $K$, i.e. of the maximal order ${𝒪}_K$ of $K$, contains positive rational integers.

Proof.  Let  $𝔞\ne (0)$  be any ideal of ${𝒪}_K$.  Take a nonzero element $\alpha$ of $𝔞$.  The norm (http://planetmath.org/NormInNumberField) of $\alpha$ is the product

$$\text{N}(\alpha )={\alpha }^{(1)}\underset{\gamma }{\underbrace{{\alpha }^{(2)}\mathrm{\cdots }{\alpha }^{(n)}}}$$

where $n$ is the degree of the number field and ${\alpha }^{(1)},{\alpha }^{(2)},\mathrm{\cdots },{\alpha }^{(n)}$ is the set of the http://planetmath.org/node/12046$\mathrm{K}$-conjugates of  $\alpha ={\alpha }^{(1)}$.  The number

$$\gamma =\frac{\text{N}(\alpha )}{\alpha }$$

belongs to the field $K$ and it is an algebraic integer, since ${\alpha }^{(2)},\mathrm{\cdots },{\alpha }^{(n)}$ are, as algebraic conjugates of $\alpha$, also algebraic integers.  Thus  $\gamma \in {𝒪}_K$.  Consequently, the non-zero integer

$$\text{N}(\alpha )=\alpha \gamma$$

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