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The ruled residue theorem characterises residue field extensions forvaluations on a rational function field. Under the assumption that thecharacteristic of the residue field is different from $2$ this theorem isextended here to function fields of conics. The main result is that there is atmost one extension of a valuation from on the base field to the function fieldof a conic for which the residue field extension is transcendental but notruled. Furthermore the situation when this valuation is present ischaracterised.

A ruled residue theorem for function fields of conics