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单词 DiscriminantInAlgebraicNumberField
释义

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.  Letϑ1=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

αi(j)=ri(ϑj).

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

Δ(α1,α2,,αn):=det(ri(ϑj))2=det(αi(j))2,

i.e.

Δ(α1,α2,,αn):=|r1(ϑ1)r1(ϑ2)r1(ϑn)r2(ϑ1)r2(ϑ2)r2(ϑn)rn(ϑ1)rn(ϑ2)rn(ϑn)|2=|α1(1)α1(2)α1(n)α2(1)α2(2)α2(n)αn(1)αn(2)αn(n)|2.

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

Δ(α1,α2,,αn)=|S(α1α1)S(α1α2)S(α1αn)S(α2α1)S(α2α2)S(α2αn)S(αnα1)S(αnα2)S(αnαn)|,

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).

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更新时间:2025/5/4 7:28:58