The polynomial x2+5x+3 in R[x]
induces the function f:R¨R defined
by the rule: f(a) = a2+5a+3 for
each a in R. This assignment defines an
surjective ring homomorphism R[x] ¨R[a] = ring
of polynomials in the variable a.
Remark: There is a distinction between polynomials and polynomial
functions (these rings are not isomorphic in general).
Consider the unequal polynomials x4+x+1 and x3+x2+1 in
the induced functions
f: Z3¨Z3 , f(a)= a4+a+1 for
each a in Z3
g: Z3¨Z3, g(a)=a3+a2+1 for each
a in Z3
Therefore, f=g, but the
polynomials that induced these functions are not equal, (i.e., the above
homomorphism is not always injective).
Note: The existence of such an example is clear,
since the polynomial ring Z3[x] is
infinite, but there are only a finite number of functions f: Z3¨Z3.
Question: Is there a similar example if the
coefficient field is infinite?
Answer: NO!!! R[x] isomorphic
to R[a] iff R is infinite.