Problem Statement: (a) State the division algorithm for (b) Show that is a principle ideal domain.
(a) The Division Algorithm: If with then there exist such that where .
(b) Proof: Let be an ideal of . If is the trivial ideal then it is principal. So suppose that is not the trivial ideal. Then there is some non-zero polynomial such that has the least degree of any polynomial in .
Let , then by the division algorithm we may write
where and . Now, since is an ideal and it follows that , so . But which is the least possible degree in , so it must be that . And so we have shown that there is a for every such that , thus, and so is a Principal Ideal Domain.
Reflection: They key in this proof is that we chose our to have the least possible degree. This makes the polynomial act kind of like a gcd does for a list of numbers.