Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T22:39:02.899Z Has data issue: false hasContentIssue false

On rational-derived quartics

Published online by Cambridge University Press:  17 April 2009

R.H. Buchholz
Affiliation:
Department of Defence Po Box 4924 Kingston, ACT 2604Australia e-mail: [email protected]
S.M. Kelly
Affiliation:
Department of Defence Po Box 4924 Kingston, ACT 2604Australia e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present a characterisation of all quartic polynomials with exactly three distinct roots and the property that it and all its derivatives have rational roots. It turns out that there are an infinite number of distinct such quartics, each of which corresponds to a point on a related elliptic curve. Furthermore the collection of these points forms a proper subgroup of the group of rational points on the curve.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Caldwell, C.K., Nice polynomials of degree 4, Mathematical Spectrum, 1989.Google Scholar
[2]Carroll, C.E., ‘Polynomials all of whose derivatives have integer roots’, Amer. Math. Monthly 26 (1989), 129130.CrossRefGoogle Scholar
[3]Chapple, M., ‘A cubic equation with rational roots such that its derived equation also has rational roots’, A Mathematics Bulletin for Teachers in Secondary Schools 11 (1960), 57.Google Scholar
[4]Galvin, B., “‘‘Nice’ cubic polynomials with ‘nice’ derivatives’, in Australian Senior Mathematics Journal 4, 1990, pp. 1721.Google Scholar
[5]Calvin, B. and MacDougall, J., Private Communication.Google Scholar
[6]Mordell, L.J., Diophantine equations (Academic Press, New York, 1969).Google Scholar
[7]Silverman, J.H., The arithmetic of elliptic curves (Springer-Verlag, Berlin, Heidelberg, New York, 1986).CrossRefGoogle Scholar
[8]Silverman, J.H. and Tate, J., Rational points on elliptic curves (Springer-Verlag, New York, 1992).CrossRefGoogle Scholar