If we take to be the ring of polynomials with integer coefficients, and to be the subset .
So I is the ring of polynomials with even integer coefficients
So far, I have shown that I is an ideal.
I am trying to
(i) find & describe the number of cosets in
(ii) Prove that is not a principle ideal of
(iii) Decide if is a prime ideal.
Well, for (i), I am thinking there is infinitely many cosets? As long as any has coefficient 1, it forms it's own coset... for example
etc... in terms of coset equality... That seems to be it to me...
, which is true...Note that an element
in the quotient ring is uniquely determined by its free coefficient's parity...
(ii) I am not quite too sure how to go about this. If we let for some and derive a contradiction (show we can't just take multiples of p?)
First show that , and then suppose . Check that this
is impossible by noting that both and checking the possible degree of
(iii) I think this will be easy once (i) is figured out... If is an integral domain, then is prime. If is not integral domain, then is not prime.
Indeed, this follows at once from (i)
Any help appreciated! Thanks in advance!