discriminant in algebraic number field

Let us consider the elements α1,α2,,αn of an algebraic number fieldMathworldPlanetmath (ϑ) of degree (http://planetmath.org/NumberField) n.  Let ϑ1=ϑ,ϑ2,,ϑn be the algebraic conjugates of the primitive elementMathworldPlanetmathPlanetmath ϑ and

αi=ri(ϑ)(i= 1, 2,,n)

the canonical forms of the elements αi.  Then the (ϑ)-conjugatesPlanetmathPlanetmath (http://planetmath.org/CharacteristicPolynomialOfAlgebraicNumber) of those elements are


Using these, one can define the discriminantMathworldPlanetmathPlanetmathPlanetmathΔ(α1,α2,,αn) of the elenents αi as




Basing on the properties of determinantsMathworldPlanetmath, one sees at once that the discriminant is of the numbers αi.  The entry independence of characteristic polynomialMathworldPlanetmathPlanetmath on primitive element allows to see that the discriminant also does not depend on the used primitive element of the field.  Moreover, the method for multiplying the determinants enables to convert the discriminant into the form


where S is the trace function defined in (ϑ); therefore the discriminant is always a rational number (and an integer if every αi is an algebraic integerMathworldPlanetmath of the field).  Cf. the parent entry (http://planetmath.org/DiscriminantOfANumberField).

Title discriminant in algebraic number field
Canonical name DiscriminantInAlgebraicNumberField
Date of creation 2013-03-22 19:09:23
Last modified on 2013-03-22 19:09:23
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Definition
Classification msc 11R29
Synonym discriminant in terms of conjugates
Related topic IndependenceOfCharacteristicPolynomialOnPrimitiveElement
Defines discriminant