# 4.5 Symmetric Polynomials

1. The units in R[x

_{1}, . . ., x_{n}] are just the units in R. If n = 1 this is Theorem 2 §4.1. In general, the units in R[x_{1}, . . ., x_{n}] = R[x_{1}, . . ., x_{n−1}][x_{n}] are the units in R[x_{1}, . . ., x_{n−1}], so it follows by induction.2. a.

3. f(x, y) = x + y + xy is symmetric but not homogeneous.

f(x, y) = x

^{2}y is homogeneous but not symmetric.5. Given θ : R → S, any homomorphism with these properties must be given by

because the c

_{i}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 ...Get *Introduction to Abstract Algebra, Solutions Manual, 4th Edition* now with O’Reilly online learning.

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