On the norm and multiplication principles for norm varieties

Let p be a prime, and suppose that F is a field of characteristic zero which is pspecial (that is, every finite field extension of F has dimension a power of p). Let α ε K^M _n (F)=p be a nonzero symbol and X/F a norm variety for α. We show that X has a K^M _m -norm principle for any m, extending the known K^M ₁ -norm principle. As a corollary we get an improved description of the kernel of multiplication by a symbol. We also give a new proof for the norm principle for division algebras over p-special fields by proving a decomposition theorem for polynomials over F-central division algebras. Finally, for p = n = m = 2 we show that the known K^M ₁ -multiplication principle cannot be extended to a K^M ₂ -multiplication principle for X.