Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T16:22:17.893Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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