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.
2. a. img
3. f(x, y) = x + y + xy is symmetric but not homogeneous.
f(x, y) = x2y is homogeneous but not symmetric.
5. Given θ : RS, any homomorphism img with these properties must be given by

img

because the ci are central in S, so img is unique if it exists. But this formula defines a map RS because the coefficients img are uniquely determined by the polynomial. Then it is routine to verify that img is a homomorphism such that img for all a img R and for all ...

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