discriminant in algebraic number field

Let us consider the elements $\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{n}$ of an algebraic number field $\mathbb{Q}(\vartheta)$ of degree (http://planetmath.org/NumberField) $n$ . Let $\vartheta_{1}=\vartheta,\,\vartheta_{2},\,\ldots,\,\vartheta_{n}$ be the algebraic conjugates of the primitive element $\vartheta$ and

\alpha_{i}\;=\;r_{i}(\vartheta)\quad(i\;=\;1,\,2,\,\ldots,\,n)

the canonical forms of the elements $\alpha_{i}$ . Then the $\mathbb{Q}(\vartheta)$ -conjugates (http://planetmath.org/CharacteristicPolynomialOfAlgebraicNumber) of those elements are

\alpha_{i}^{(j)}\;=\;r_{i}(\vartheta_{j}).

Using these, one can define the discriminant $\Delta(\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{n})$ of the elenents $\alpha_{i}$ as

\Delta(\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{n})\;:=\;\det(r_{i}(% \vartheta_{j}))^{2}\;=\;\det\!\left(\alpha_{i}^{(j)}\right)^{2}\!,

i.e.

\Delta(\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{n})\;:=\;\left|\begin{array}% []{cccc}r_{1}(\vartheta_{1})&r_{1}(\vartheta_{2})&\ldots&r_{1}(\vartheta_{n})% \\ r_{2}(\vartheta_{1})&r_{2}(\vartheta_{2})&\ldots&r_{2}(\vartheta_{n})\\ \vdots&\vdots&\ddots&\vdots\\ r_{n}(\vartheta_{1})&r_{n}(\vartheta_{2})&\ldots&r_{n}(\vartheta_{n})\end{% array}\right|^{2}\;=\;\left|\begin{array}[]{cccc}\alpha_{1}^{(1)}&\alpha_{1}^{% (2)}&\ldots&\alpha_{1}^{(n)}\\ \alpha_{2}^{(1)}&\alpha_{2}^{(2)}&\ldots&\alpha_{2}^{(n)}\\ \vdots&\vdots&\ddots&\vdots\\ \alpha_{n}^{(1)}&\alpha_{n}^{(2)}&\ldots&\alpha_{n}^{(n)}\end{array}\right|^{2% }\!.

Basing on the properties of determinants, one sees at once that the discriminant is of the numbers $\alpha_{i}$ . The entry independence of characteristic polynomial 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

\Delta(\alpha_{1},\,\alpha_{2},\,\ldots,\,\alpha_{n})\;=\;\left|\begin{array}[% ]{cccc}\mbox{S}(\alpha_{1}\alpha_{1})&\mbox{S}(\alpha_{1}\alpha_{2})&\ldots&% \mbox{S}(\alpha_{1}\alpha_{n})\\ \mbox{S}(\alpha_{2}\alpha_{1})&\mbox{S}(\alpha_{2}\alpha_{2})&\ldots&\mbox{S}(% \alpha_{2}\alpha_{n})\\ \vdots&\vdots&\ddots&\vdots\\ \mbox{S}(\alpha_{n}\alpha_{1})&\mbox{S}(\alpha_{n}\alpha_{2})&\ldots&\mbox{S}(% \alpha_{n}\alpha_{n})\end{array}\right|\!,

where S is the trace function defined in $\mathbb{Q}(\vartheta)$ ; therefore the discriminant is always a rational number (and an integer if every $\alpha_{i}$ is an algebraic integer 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