Simply, it is a method for transforming a polynomial equation of degree with some nonzero intermediate coefficients, , such that some or all of the transformed intermediate coefficients, , are exactly zero.
where is modulo . That is, any element of is a polynomial in , which is thus a primitive element of . There will be other choices of primitive element in : for any such choice of we will have by definition:
,
with polynomials and over . Now if is the minimal polynomial for over , we can call a Tschirnhaus transformation of .
Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing , but leaving the same. This concept is used in reducing quintics to Bring–Jerrard form, for example. There is a connection with Galois theory, when is a Galois extension of . The Galois group may then be considered as all the Tschirnhaus transformations of to itself.
Historyedit
In 1683, Ehrenfried Walther von Tschirnhaus published a method for rewriting a polynomial of degree such that the and terms have zero coefficients. In his paper, Tschirnhaus referenced a method by René Descartes to reduce a quadratic polynomial such that the term has zero coefficient.
In 1786, this work was expanded by Erland Samuel Bring who showed that any generic quintic polynomial could be similarly reduced.
In 1834, George Jerrard further expanded Tschirnhaus' work by showing a Tschirnhaus transformation may be used to eliminate the , , and for a general polynomial of degree .[3]
^ abvon Tschirnhaus, Ehrenfried Walter; Green, R. F. (2003-03-01). "A method for removing all intermediate terms from a given equation". ACM SIGSAM Bulletin. 37 (1): 1–3. doi:10.1145/844076.844078. ISSN 0163-5824. S2CID 18911887.
^Garver, Raymond (1927). "The Tschirnhaus Transformation". Annals of Mathematics. 29 (1/4): 319–333. doi:10.2307/1968002. ISSN 0003-486X. JSTOR 1968002.
^ abC. B. Boyer (1968) A History of Mathematics. Wiley, New York pp. 472-473. As reported by: Weisstein, Eric W. "Tschirnhausen Transformation". mathworld.wolfram.com. Retrieved 2022-02-02.
March 13, 2024
tschirnhaus, transformation, mathematics, also, known, tschirnhausen, transformation, type, mapping, polynomials, developed, ehrenfried, walther, tschirnhaus, 1683, ehrenfried, walther, tschirnhaussimply, method, transforming, polynomial, equation, degree, dis. 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 1 Ehrenfried Walther von TschirnhausSimply it is a method for transforming a polynomial equation of degree n 2 displaystyle n geq 2 with some nonzero intermediate coefficients a 1 a n 1 displaystyle a 1 a n 1 such that some or all of the transformed intermediate coefficients a 1 a n 1 displaystyle a 1 a n 1 are exactly zero For example finding a substitutiony x k 1 x 2 k 2 x k 3 displaystyle y x k 1 x 2 k 2 x k 3 for a cubic equation of degree n 3 displaystyle n 3 f x x 3 a 2 x 2 a 1 x a 0 displaystyle f x x 3 a 2 x 2 a 1 x a 0 such that substituting x x y displaystyle x x y yields a new equationf y y 3 a 2 y 2 a 1 y a 0 displaystyle f y y 3 a 2 y 2 a 1 y a 0 such that a 1 0 displaystyle a 1 0 a 2 0 displaystyle a 2 0 or both More generally 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 Contents 1 Definition 2 Example Tschirnhaus method for cubic equations 3 Generalization 4 History 5 See also 6 ReferencesDefinition editFor a generic n t h displaystyle n th nbsp degree reducible monic polynomial equation f x 0 displaystyle f x 0 nbsp of the form f x g x h x displaystyle f x g x h x nbsp where g x displaystyle g x nbsp and h x displaystyle h x nbsp are polynomials and h x displaystyle h x nbsp does not vanish at f x 0 displaystyle f x 0 nbsp f x x n a 1 x n 1 a 2 x n 2 a n 1 x a n 0 displaystyle f x x n a 1 x n 1 a 2 x n 2 a n 1 x a n 0 nbsp the Tschirnhaus transformation is the function y k 1 x n 1 k 2 x n 2 k n 1 x k n displaystyle y k 1 x n 1 k 2 x n 2 k n 1 x k n nbsp Such that the new equation in y displaystyle y nbsp f y displaystyle f y nbsp has certain special properties most commonly such that some coefficients a 1 a n 1 displaystyle a 1 a n 1 nbsp are identically zero 2 3 Example Tschirnhaus method for cubic equations editIn Tschirnhaus 1683 paper 1 he solved the equationf x x 3 p x 2 q x r 0 displaystyle f x x 3 px 2 qx r 0 nbsp using the Tschirnhaus transformation y x a x a x y a x y a displaystyle y x a x a longleftrightarrow x y a x y a nbsp Substituting yields the transformed equationf y a y 3 3 a p y 2 3 a 2 2 p a q y a 3 p a 2 q a r 0 displaystyle f y a y 3 3a p y 2 3a 2 2pa q y a 3 pa 2 qa r 0 nbsp or a 1 3 a p a 2 3 a 2 2 p a q a 3 a 3 p a 2 q a r displaystyle begin cases a 1 3a p a 2 3a 2 2pa q a 3 a 3 pa 2 qa r end cases nbsp Setting a 1 0 displaystyle a 1 0 nbsp yields 3 a p 0 a p 3 displaystyle 3a p 0 rightarrow a frac p 3 nbsp and finally the Tschirnhaus transformation y x p 3 displaystyle y x frac p 3 nbsp which may be substituted into f y a displaystyle f y a nbsp to yield an equation of the form f y y 3 q y r displaystyle f y y 3 q y r nbsp Tschirnhaus went on to describe how a Tschirnhaus transformation of the form x 2 y a b x 2 b x y a displaystyle x 2 y a b x 2 bx y a nbsp may be used to eliminate two coefficients in a similar way Generalization editIn detail let K displaystyle K nbsp be a field and P t displaystyle P t nbsp a polynomial over K displaystyle K nbsp If P displaystyle P nbsp is irreducible then the quotient ring of the polynomial ring K t displaystyle K t nbsp by the principal ideal generated by P displaystyle P nbsp K t P t L displaystyle K t P t L nbsp is a field extension of K displaystyle K nbsp We have L K a displaystyle L K alpha nbsp where a displaystyle alpha nbsp is t displaystyle t nbsp modulo P displaystyle P nbsp That is any element of L displaystyle L nbsp is a polynomial in a displaystyle alpha nbsp which is thus a primitive element of L displaystyle L nbsp There will be other choices b displaystyle beta nbsp of primitive element in L displaystyle L nbsp for any such choice of b displaystyle beta nbsp we will have by definition b F a a G b displaystyle beta F alpha alpha G beta nbsp with polynomials F displaystyle F nbsp and G displaystyle G nbsp over K displaystyle K nbsp Now if Q displaystyle Q nbsp is the minimal polynomial for b displaystyle beta nbsp over K displaystyle K nbsp we can call Q displaystyle Q nbsp a Tschirnhaus transformation of P displaystyle P nbsp Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing P displaystyle P nbsp but leaving L displaystyle L nbsp the same This concept is used in reducing quintics to Bring Jerrard form for example There is a connection with Galois theory when L displaystyle L nbsp is a Galois extension of K displaystyle K nbsp The Galois group may then be considered as all the Tschirnhaus transformations of P displaystyle P nbsp to itself History editIn 1683 Ehrenfried Walther von Tschirnhaus published a method for rewriting a polynomial of degree n gt 2 displaystyle n gt 2 nbsp such that the x n 1 displaystyle x n 1 nbsp and x n 2 displaystyle x n 2 nbsp terms have zero coefficients In his paper Tschirnhaus referenced a method by Rene Descartes to reduce a quadratic polynomial n 2 displaystyle n 2 nbsp such that the x displaystyle x nbsp term has zero coefficient In 1786 this work was expanded by Erland Samuel Bring who showed that any generic quintic polynomial could be similarly reduced In 1834 George Jerrard further expanded Tschirnhaus work by showing a Tschirnhaus transformation may be used to eliminate the x n 1 displaystyle x n 1 nbsp x n 2 displaystyle x n 2 nbsp and x n 3 displaystyle x n 3 nbsp for a general polynomial of degree n gt 3 displaystyle n gt 3 nbsp 3 See also editPolynomial transformations Monic polynomial Reducible polynomial Quintic function Galois theory Abel Ruffini theoremReferences edit a b von Tschirnhaus Ehrenfried Walter Green R F 2003 03 01 A method for removing all intermediate terms from a given equation ACM SIGSAM Bulletin 37 1 1 3 doi 10 1145 844076 844078 ISSN 0163 5824 S2CID 18911887 Garver Raymond 1927 The Tschirnhaus Transformation Annals of Mathematics 29 1 4 319 333 doi 10 2307 1968002 ISSN 0003 486X JSTOR 1968002 a b C B Boyer 1968 A History of Mathematics Wiley New York pp 472 473 As reported by Weisstein Eric W Tschirnhausen Transformation mathworld wolfram com Retrieved 2022 02 02 Retrieved from https en wikipedia org w index php title Tschirnhaus transformation amp oldid 1204720944, wikipedia, wiki, book, books, library,