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AbstractWe prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.doi:10.2478/s11533-013-0218-8 fatcat:y4yra5s57ne3nmhtvy7goict5a