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A note on Ohm's rationality criterion for conics
Published online by Cambridge University Press: 17 April 2009
Abstract
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
- Information
- Bulletin of the Australian Mathematical Society , Volume 51 , Issue 1 , February 1995 , pp. 133 - 137
- Copyright
- Copyright © Australian Mathematical Society 1995