4. Let K be any field, and let S nomial ring in the m indeterminates over K. K[X1, X2,. Xm] denote the poly- (1) For any (a,..., am) E KM, let Ma....am be the ideal (X-a,..., Xm - am) of S. Show that Ma,..., am is a maximal ideal, and that Ma,..., am if and only if f(a,..., am) 0. = (2) Let R = K[Y, ... , Yn], and define the K-algebra homomorphismo: R S as the unique homomorphism such that o(Y;) Fi(X,..., Xm) for i 1,..., n. Show that x = a, ... , xm = am is a simultaneous solution of the equations = - F(x,...,xm) = b, for i = = 1, n if and only if M... Ma,..., am, where M...bn is the extension to S of the ideal Mb,...,bn = (Y - b, ..., Yn - bn) of R.