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A note on Ohm's rationality criterion for conics

Published online by Cambridge University Press:  17 April 2009

Mowaffaq Hajja
Affiliation:
Mathematics DepartmentYarmouk UniversityIrbid, Jordan
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Abstract

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An irreducible quadratic polynomial P(X, Y) in two variables over a field k is called a conic over k. It is called rational if its function field is simple transcendental over k (equivalently if P is parameterisable by rational functions). Ohm's rationality criterion states that P is rational if and only if (i) the locus of P is non-empty and (ii) k is algebraically closed in the function field of P. To show the irredundancy of (ii), Ohm gives an example of a non-rational conic with a non-empty locus. That locus, however, consists of a single point.

In this note, we show that a better example cannot exist by showing that if the locus of a conic contains more than one point then it is rational. We also show that the only rational conic whose locus consists of one point is the conic XY + 1 over the field of two elements.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Ohm, J., ‘Function fields of conics, a theorem of Amitsur-MacRae, and a problem of Zariski’, in Algebraic Geometry and its applications, (Bajaj, C., Editor) (Springer-Verlag, Berlin, Heidelberg, New York, 1993), pp. 351382.Google Scholar