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The quartic equation: invariants and Euler's solution revealed

Published online by Cambridge University Press:  01 August 2016

R. W. D. Nickalls
R. W. D. Nickalls*
Affiliation:
Department of Anaesthesia, Nottingham University Hospitals, City Hospital Campus, Nottingham NG5 1PB, UKe-mail: dick@nickalls.org
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Extract

The central role of the resolvent cubic in the solution of the quartic was first appreciated by Leonard Euler (1707-1783). Euler's quartic solution first appeared as a brief section (§ 5) in a paper on roots of equations [1, 2], and was later expanded into a chapter entitled ‘Of a new method of resolving equations of the fourth degree’ (§§ 773-783) in his Elements of algebra [3,4].

Type
Articles
Information
The Mathematical Gazette , Volume 93 , Issue 526 , March 2009 , pp. 66 - 75
Copyright
Copyright © The Mathematical Association 2009

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References

1. Euler, L., De formis radicum aequationum cujusque ordinis conjectatio. Commentarii academiae scientiarum imperialis Petropolitianae 6 (1733), pp. 216–231 = Opera Omnia, Series 1, 6 (Theory of equations) pp. 1–19. [Euler Archive, E30 (Latin): http://math.dartmouth.edu/~euler/] Google Scholar
2. Bell, J., A conjecture on the forms of the roots of equations. arA7v:0806.1927vl [math.HO] (2008). http://arxiv.org/abs/0806.1927 [An English translation of Euler’s De formis radicum aequationum cujusque ordinis conjectatio (E30)].Google Scholar
3. Euler, L., Vollständige Anleitung zur Algebra (Elements of algebra), 2 vols, Royal Academy of Sciences, St. Petersburg (1770). [Euler Archive, E387 (English): http://math.dartmouth.edu/~euler/ docs/originals/E387e.PlS4.pdf] Google Scholar
4. Sangwin, C. R. (ed.). Euler’s Elements of algebra, Tarquin Publications, St Albans, UK (2006). [English translation of Euler 1770 (E387)]Google Scholar
5. Burnside, W. S. and Panton, A. W.. The theory of equations: with an introduction to the theory of binary algebraic forms (7th edn.), 2 vols; Longmans, Green and Co., London (1912).Google Scholar
6. Cremona, J. E.. Reduction of binary cubic and quartic forms, J. Comput. Math. 2 (1999) pp. 62–92. www.lms.ac.uk/jcm/2/ lms98007/ [the seminvariants G and H2 – 16a2/ are denoted here by R and 3Q respectively]Google Scholar
7. Olver, P. J., Classical invariant theory, London Mathematical Society Student Texts No. 44, Cambridge University Press (1999).CrossRefGoogle Scholar
8. Nickalls, R. W. D., A new approach to solving the cubic: Cardan’s solution revealed. Math. Gaz., 77 (November 1993) pp. 354–359. www.nickalls.org/dick/papers/maths/nickallscubic/1993.pdf CrossRefGoogle Scholar
9. Ball, R. S., Note on the algebraical solution of biquadratic equations, Quarterly Journal of Pure and Applied Mathematics 7 (1866) pp. 6–9, 358–369.Google Scholar
10. Dalton, J. P., On the graphical discrimination of the cubic and of the quartic, Math. Gaz. 17 (July 1933) pp. 189–196.CrossRefGoogle Scholar