Exercise 3.

Let be a non-constant polynomial of degree . Let be the set of the residue classes modulo .
1.
Let be a polynomial. How many elements are there in the residue class of modulo with degree less than ?
2.
Explain why we can view each element of as a vector of coordinates in .
Given two polynomial and in we want to decide whether there exists a polynomial such that we have . For instance, with , and , one solution is . Indeed, we have . (Working modulo means replacing every occurrence of by .)
3.
Assume first that . Explain why such a polynomial exists and is unique.
4.
Now, assume that where is a non-constant polynomial. Indicate briefly a strategy for solving our problem in this case.
5.
Solve our problem for the particular , and .

Marc Moreno Maza
2008-01-31