explicit generators of a quotient polynomial ring associated to a given polynomial
Let be a field and consider ring of polynomials . If and , then we will write to denote element in represented by .
Lemma. Assume, that are different elements and . Let be given by . Then there exist such that divides polynomial
Proof. Note, that for any . Thus we may define . Then for any we have , therefore
is such that for any . In particular and thus divides for any . This completes the proof.
Corollary. Under the same assumptions as in lemma, we have that ideal in is equal to .
Proof. Indeed, all we need to show is that we can generate . Lemma implies, that there is such that
This completes the proof.
Remark. This gives us an explicit formula for generators of . In particular the dimension over of this ring is at most . It can be shown that actualy it is equal, even if is arbitrary.
|Title||explicit generators of a quotient polynomial ring associated to a given polynomial|
|Date of creation||2013-03-22 19:10:01|
|Last modified on||2013-03-22 19:10:01|
|Last modified by||joking (16130)|