The
polynomial *x*2+5*x*+3 in **R**[*x*]
induces the function *f*:**R**¨**R** defined

by the rule: *f*(*a*) = *a*2+5*a*+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).

**Example:** Consider the
__unequal__ polynomials *x*4+*x*+1 and *x*3+*x*2+1 in

**Z**3[*x*] and the induced functions

*f*: **Z**3¨**Z**3 , *f*(*a*)= *a*4+*a*+1 for
each *a* in **Z**3

*g*: **Z**3¨**Z**3, *g*(*a*)=*a*3+*a*2+1 for each
*a* in **Z**3

*f*(0)=1=*g*(0); *f*(1)=3=0,
*g*(1)=3=0;
*f*(2)=19=1,
*g*(2)=13=1

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 **Z**3[*x*] is infinite, but there are only a
finite number of functions *f*: **Z**3¨**Z**3.

**Question:
**Is there a similar example** **if the
coefficient field is infinite?

**Answer: NO!!! ****R**[*x*] isomorphic
to **R**[*a*] iff R is infinite.