A version of the Hardy–Littlewood circle method is developed for number fields and is used to show that nonsingular projective cubic hypersurfaces over always have a -rational point when they have dimension at least .
T. D. Browning. P. Vishe. "Cubic hypersurfaces and a version of the circle method for number fields." Duke Math. J. 163 (10) 1825 - 1883, 15 July 2014. https://doi.org/10.1215/00127094-2738530