- #1

Bashyboy

- 1,421

- 5

## Homework Statement

Give an example of a commutative ring ##R## and ##f(x), g(x) R[x]## with ##f## monic such that the remainder after dividing ##g## by ##f## is not unique; that is, there are ##q,q',r,r' \in R[x]## with ##qf + r = g = q' f + r'## and ##\deg (r)## and ##\deg (r')## are both strictly less than ##\deg(f)##.

## Homework Equations

## The Attempt at a Solution

Okay. I have been thinking about this problem for a rather long time ---I've been foiling for hours (perhaps an exaggeration). I have tried several pairs of polynomials when the base ring is ##\Bbb{Z}_4##, ##\Bbb{Z}_6##, and ##\Bbb{Z}_8##. My strategy was to solve for ##q## when ##r=0##; and then find a ##q'## given some nonzero ##r'##. I set up with equations with five variables so that I could find a nonzero remainder with ##\deg (r) < \deg (f)##, but I could find no values for these five variables. I could use a hint on finding a ring ##R## and polynomials ##f(x)## and ##g(x)##. But please don't just hand them to me; I would like to figure out this problem on my own as much as possible.