Let be a non-negative integer. We prove that a perfect field has cohomological dimension at most if, and only if, for any finite extension of and for any homogeneous space under a smooth linear connected algebraic group over , the -th Milnor -theory group of is spanned by the images of the norms coming from finite extensions of over which has a rational point. We also prove a variant of this result for imperfect fields.
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Diego Izquierdo, Giancarlo Lucchini Arteche, Homogeneous spaces, algebraic -theory and cohomological dimension of fields. J. Eur. Math. Soc. 24 (2022), no. 6, pp. 2169–2189DOI 10.4171/JEMS/1129