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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, UK e-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].

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Articles
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The Mathematical Gazette , Volume 93 , Issue 526 , March 2009, pp. 66 - 75
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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
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