4.5 Symmetric Polynomials
1. The units in R[x1, . . ., xn] are just the units in R. If n = 1 this is Theorem 2 §4.1. In general, the units in R[x1, . . ., xn] = R[x1, . . ., xn−1][xn] are the units in R[x1, . . ., xn−1], so it follows by induction.
3. f(x, y) = x + y + xy is symmetric but not homogeneous.
f(x, y) = x2y is homogeneous but not symmetric.
5. Given θ : R → S, any homomorphism with these properties must be given by
because the ci are central in S, so is unique if it exists. But this formula defines a map R → S because the coefficients are uniquely determined by the polynomial. Then it is routine to verify that is a homomorphism such that for all a R and for all ...