International Congress of Mathematicians

82 Markus Rost4. Hubert's 90 for symbolsThe bijectivity of the norm residue homomorphisms has always been consideredas a sort of higher version of the classical Hilbert's Theorem 90 (which establishesthe bijectivity for n = 1). In fact, there are various variants of the Bloch-Katoconjecture which are obvious generalizations of Hilbert's Theorem 90: The Hilbert'sTheorem 90 for Kff of cyclic extensions or the vanishing of the motivic cohomologygroup H n+1 (k, Z(n)). In this section we describe a variant which on one hand isvery elementary to formulate and on the other hand is the really hard part of theBloch-Kato conjecture (modulo Voevodsky's theorem).Let u = {cti,... ,a n } £ Kffh/p be a symbol. Consider the norm mapA4 = ^2 N F/k • 0FFKiF -• Kikwhere F runs through the finite field extensions of k (contained in some algebraicclosure of k) which split u. Hilbert's Theorem 90 for u states that ker A4 is generatedby the "obvious" elements.To make this precise, we consider two types of basic relations between thenorm maps N F / k .Let Fi, F 2 be finite field extensions of k. Then the sequenceir IT? o,r\ ( N F 1 IS,F 2 /F 1 ,-N FIIS!F2/F2 ) N Fl/k +Np 2/kKi(Li®i