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Let k ⊆ k′ be a field extension. We give relations between the kernels of higher derivations on k[ X] and k′[ X], where k[ X]:= k[ x1,…, xn] denotes the polynomial ring in n variables over the field k. More precisely, let D = { Dn} n=0∞ a higher k-derivation on k[ X] and D′ = { D′ n} n=0∞ a higher k′-derivation on k′[ X] such that D′ m( xi) = Dm( xi) for all m ⩾ 0 and i = 1, 2,…, n. Then (1) k[ X] D = k if and only if k′[ X] D′ = k′; (2) k[ X] D is a finitely generated k-algebra if and only if k′[ X] D′ is a finitely generated k′-algebra. Furthermore, we also show that the kernel k[ X] D of a higher derivation D of k[ X] can be generated by a set of closed polynomials. [ABSTRACT FROM AUTHOR]

The two-dimensional local field K = F q(( u))(( t)), char K = p, and its Brauer group Br( K) are considered. It is proved that, if L = K( x) is the field extension for which we have x p − x = ut − p =: h, then the condition that ( y, f | h]K = 0 for any y ε K is equivalent to the condition f ε Im(Nm( L*)). [ABSTRACT FROM AUTHOR]