properties of discriminant in algebraic number field

Theorem 1.  Let α1,α2,,αn and β1,β2,,βn be elements of the algebraic number fieldMathworldPlanetmath (ϑ) of degree ( n.  If they satisfy the equations

αi=j=1ncijβjfori= 1, 2,,n,

where all coefficientsMathworldPlanetmath cij are rational numbersPlanetmathPlanetmathPlanetmath, then the are via the equation


As a special case one obtains the

Theorem 2.  If

αi=ci1+ci2ϑ++cinϑn-1fori= 1, 2,,n (1)

are the canonical forms of the elements αi in (ϑ), then


where Δ(1,ϑ,,ϑn-1) is a Vandermonde determinantMathworldPlanetmath thus having the product form

Δ(1,ϑ,,ϑn-1)=(1ij(ϑj-ϑi))2=1ij(ϑi-ϑj)2 (2)

where ϑ1=ϑ,ϑ2,,ϑn are the algebraic conjugates of ϑ.

Since the (2) is also the polynomial discriminant of the irreducible minimal polynomialPlanetmathPlanetmath of ϑ, the numbers ϑi are inequal.  It follows the

Theorem 3.  When (1) are the canonical forms of the numbers αi, one has

Δ(α1,α2,,αn)= 0det(cij)= 0.

The powers 1,ϑ,,ϑn-1 of the primitive elementMathworldPlanetmathPlanetmath ( form a basis ( of the field extension (ϑ)/ (see the canonical form of element of number field).  By the theorem 3 we may write the

Theorem 4.  The numbers α1,α2,,αn of (ϑ) are linearly independentMathworldPlanetmath over if and only if  Δ(α1,α2,,αn) 0.

Theorem 5.(α)=(ϑ)Δ(1,α,α2,,αn-1)0.  Here, the the discriminant is the discriminant of the algebraic numberMathworldPlanetmath ( α.

Title properties of discriminant in algebraic number field
Canonical name PropertiesOfDiscriminantInAlgebraicNumberField
Date of creation 2013-03-22 19:09:28
Last modified on 2013-03-22 19:09:28
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Result
Classification msc 11R29
Related topic MinimalityOfIntegralBasis
Related topic ConditionForPowerBasis