An estimate for character sums

It turns out to be easier to consider the following more general situation. F is a finite field, n ~ 1 is an integer, and B is a finite etale F -algebra of dimension n over F (i,e., over a finite extension K of F, there exists an isomorphism of K-algebras B ®F K:.::= K x K x··· x K). We assume given an element x in B that is regular in the sense that its characteristic polynomial detF(T x I B) in the regular representation of B on itself has n distinct eigenvalues. (In terms of the above isomorphism B ® F K :.::= K x K x ... x K, x is regular if and only if x ® 1 :.::= (XI' , •.• x n ) with all distinct components x j • Or equivalently, x is regular if and only if B is equal to the F-subalgebra F[x] generated by x. In the special case when B is a field F , the element x is regular if and only if F(x) = E.)