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. 
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
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 ...
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 the O’Reilly learning platform.
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