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- Tschirnhaus_transformation abstract "In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. It may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a different choice of primitive element. This is the most general transformation of an irreducible polynomial that takes a root to some rational function applied to that root.In detail, let K be a field, and P(t) a polynomial over K. If P is irreducible, thenK[t]/(P(t)) = L,the quotient ring of the polynomial ring K[t] by the principal ideal generated by P, is a field extension of K. We have L = K(α)where α is t modulo (P). That is, α is a primitive element of L. There will be other choices β of primitive element in L: for any such choice of β we will have β = F(α), α = G(β), with polynomials F and G over K. In fact this follows from the quotient representation above. Now if Q is the minimal polynomial for β over K, we can call Q a Tschirnhaus transformation of P. Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing P, but leaving L the same. This concept is used in reducing quintics to Bring–Jerrard form, for example. There is a connection with Galois theory, when L is a Galois extension of K. The Galois group is then described (in one way) as all the Tschirnhaus transformations of P to itself.".
- Tschirnhaus_transformation wikiPageExternalLink tschirnhaus.pdf.
- Tschirnhaus_transformation wikiPageID "595824".
- Tschirnhaus_transformation wikiPageRevisionID "591761415".
- Tschirnhaus_transformation hasPhotoCollection Tschirnhaus_transformation.
- Tschirnhaus_transformation title "Tschirnhausen Transformation".
- Tschirnhaus_transformation urlname "TschirnhausenTransformation".
- Tschirnhaus_transformation subject Category:Field_theory.
- Tschirnhaus_transformation subject Category:Polynomials.
- Tschirnhaus_transformation type Abstraction100002137.
- Tschirnhaus_transformation type Function113783816.
- Tschirnhaus_transformation type MathematicalRelation113783581.
- Tschirnhaus_transformation type Polynomial105861855.
- Tschirnhaus_transformation type Polynomials.
- Tschirnhaus_transformation type Relation100031921.
- Tschirnhaus_transformation comment "In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683. It may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a different choice of primitive element.".
- Tschirnhaus_transformation label "Methode van Tschirnhaus".
- Tschirnhaus_transformation label "Méthode de Tschirnhaus".
- Tschirnhaus_transformation label "Transformación de Tschirnhaus".
- Tschirnhaus_transformation label "Tschirnhaus transformation".
- Tschirnhaus_transformation label "Tschirnhaus-Transformation".
- Tschirnhaus_transformation label "Преобразование Чирнгауза".
- Tschirnhaus_transformation sameAs Tschirnhaus-Transformation.
- Tschirnhaus_transformation sameAs Transformación_de_Tschirnhaus.
- Tschirnhaus_transformation sameAs Méthode_de_Tschirnhaus.
- Tschirnhaus_transformation sameAs Methode_van_Tschirnhaus.
- Tschirnhaus_transformation sameAs m.02tq57.
- Tschirnhaus_transformation sameAs Q2670133.
- Tschirnhaus_transformation sameAs Q2670133.
- Tschirnhaus_transformation sameAs Tschirnhaus_transformation.
- Tschirnhaus_transformation wasDerivedFrom Tschirnhaus_transformation?oldid=591761415.
- Tschirnhaus_transformation isPrimaryTopicOf Tschirnhaus_transformation.