Steenrod operations and quadratic forms

Steenrod operations on the Chow groups modulo a prime number p are not available when the characteristic of the base field is equal to p. We build operations on the restriction to a splitting field of the Chow group of a smooth projective homogeneous variety under a semi-simple linear algebraic group. These operations respect rationality of cycles provided that the base field admits a form of resolution of singularities, which is given by a result of Gabber when the base field has a characteristic different from p. Therefore we recover a weak form of Steenrod operations, in the cases when they are already constructed, using a very different approach. We show that the first Steenrod square (p = 2) can be constructed without using resolution of singularities. As a consequence we prove a theorem on the parity of the Witt index of a quadratic form. Another part of this work consists of proving directly that Chow motives of smooth projective quadrics decompose in the same way when the coefficients are either Z or Z/2.