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Picking genuine zeros of cubics in the Tschirnhaus method

Published online by Cambridge University Press:  14 March 2016

Raghavendra G. Kulkarni
Raghavendra G. Kulkarni*
Affiliation:
Department of Electronics & Communication Engineering, PES University, 100 Feet Ring Road, BSK III Stage, Bengaluru 560085, India e-mail: raghavendrakulkarni@pes.edu
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Extract

In 1683, the German mathematician Ehrenfried Walther von Tschirnhaus introduced a polynomial transformation which, he claimed, would eliminate all intermediate terms in a polynomial equation of any degree, thereby reducing it to a binomial form from which roots can easily be extracted [1]. As mathematicians at that time were struggling to solve quintic equations in radicals with no sign of any success, the Tschirnhaus transformation gave them some hope, and in 1786, Bring was able to reduce the general quintic to the form x5 + ax + b = 0, even though he didn't succeed in his primary mission of solving it. It seems Bring's work got lost in the archives of University of Lund. Unaware of Bring's work, Jerrard (1859) also arrived at the same form of the quintic using a quartic Tschirnhaus transformation [2]. From the works of Abel (1826) and Galois (1832) it is now clear that the general polynomial equation of degree higher than four cannot be solved in radicals.

Type
Research Article
Information
The Mathematical Gazette , Volume 100 , Issue 547 , March 2016 , pp. 48 - 53
Copyright
Copyright © Mathematical Association 2016 

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References

1.Tschirnhaus, Ehrenfried W., A method for removing all intermediate terms from a given equation, Ada Eruditorum, (May 1683) pp. 204–207. Translated by Green, R. F. in ACM SIGSAM Bulletin, Vol. 37, No. 1 (March 2003) pp. 1–3.Google Scholar
2.Adamchik, Victor S. and Jeffrey, David J., Polynomial transformations of Tschirnhaus, Bring and Jerrard, ACM SIGSAM Bulletin, Vol. 37, No. 3 (September 2003) pp. 90–94.Google Scholar
3.Garver, Raymond, The binomial quartic as a normal form, Bulletin of the American Mathematical Society, Vol. 33, No. 6 (November – December 1927) pp. 677–680.CrossRefGoogle Scholar
4.Cayley, A., Note on a cubic equation, Messenger of Mathematics, Vol. 15 (1886) pp. 62–64.Google Scholar
5.Burnside, W. S. and Panton, A. W., The theory of equations, Vol II, Longmans, Green (1928).Google Scholar
6.Garver, Raymond, Transformations on cubic equations, Amer. Math. Monthly, Vol. 36, No. 7 (August – September 1929) pp. 366–369.CrossRefGoogle Scholar
7.Garver, Raymond, Tschirnhaus transformations on certain rational cubics, Amer. Math. Monthly, Vol. 34, No. 10 (December 1927) pp. 521–525.Google Scholar
8.Garver, Raymond, Two applications of Tschirnhaus transformations in the elementary theory of equations, Amer. Math. Monthly, Vol. 38, No. 4 (April 1931) pp. 185–188.CrossRefGoogle Scholar
9.Deakin, Michael, History of mathematics: solving cubic equations, Parabola, Vol. 41, No. 3 (2005) pp. 1–5.Google Scholar
10.Dickson, Leonard Eugene, First course in the theory of equations, John Wiley & Sons (1922) p. 53.Google Scholar